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This article is cited in 1 scientific paper (total in 1 paper) Density of quantized approximations P. A. Borodinab, K. S. Shklyaevab a Faculty of Mechanics and Mathematics, Moscow State University b Moscow Center for Fundamental and Applied Mathematics
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     1. IntroductionThe term ’quantization’ in approximation theory (see, for example, [34] and [21]) usually means the following: the coefficients with which the elements of a given set are taken in an approximating linear combination must be proportional to a fixed number (’quantum’). If we take this permissible ’quantum’ equal to 1 and limit ourselves, firstly, to approximation in some norm and, secondly, to the level of Weierstrass’s theorem (whether or not it is possible to approximate with any accuracy), then we arrive at the following setting. Problem 1.1. Let $M$ be a given subset of a Banach space $X$. Is it true that the set is dense in $X$, that is, every element of $X$ can be approximated with arbitrary accuracy by finite sums of elements of $M$? The set $R(M)$ is the additive semigroup generated by $M$. Therefore, Problem 1.1 may be regarded as a particular case of the problem of the density of a semigroup in a Banach space (throughout, we use the term density instead of density everywhere and to be dense instead of to form an everywhere dense set). Since Problem 1.1 involves no multiplication by scalars, the space $X$ in this problem may always be assumed to be real. Formally, approximations by semigroup cover even $n$-term approximations ($M=\{\lambda d\colon \lambda\in {\mathbb R}, d\in D\}$, where $D$ is a dictionary, that is, a complete system of unit elements of $X$; see, for example, [76]), though $n$-term approximations cannot be regarded as quantized. A more explicit special case of Problem 1.1 is the rather deeply researched problem of the possibility of approximation by polynomials with integer coefficients ($M=\{\pm 1, \pm t, \pm t^2, \dots\}$, where the variable $t$ belongs to some real or complex domain; see, for instance, [77], [42] and § 5) below. Our main motivation to study Problem 1.1 comes from the theory of approximation by simple partial fractions (logarithmic derivatives of polynomials)
$$
\begin{equation*}
\sum_{k=1}^n \frac{1}{z-a_k}\,, \qquad a_k\in {\mathbb C},
\end{equation*}
\notag
$$
in various Banach spaces $X$ of functions defined on various subsets of the complex plane. Here we take $M(E)=\biggl\{\dfrac{1}{z-a}\colon a\in E\biggr\}$, $E\subset {\mathbb C}$, as the generating set, and consider the problem of density of the semigroup $\operatorname{SF}(E)=R(M(E))$ in $X$, that is, the problem of the density in $X$ of simple partial fractions with poles in $E$. This problem has a natural physical interpretation. A simple partial fraction with poles $\{a_k\}$ is complex conjugate to the intensity function of a plane electrostatic field produced by equal charges of the same sign positioned at the points $a_k$ ([60], Chap. 3, § 2; more precisely, we mean the field produced by parallel straight lines-conductors in a three-dimensional space with uniformly distributed equal charges in a plane perpendicular to these straight lines). Thus, in the problem specified an arbitrary plane electrostatic field whose intensity function belongs to the space $X$ is approximated in the norm of $X$ by a field produced by identical charges located in the set $E$. The studies of approximations by simple partial fractions began in Russia on an initiative of Dolzhenko in the early 2000s. They were perceived as a new setting in the theory of approximation of functions of a complex variable, and Russian authors obtained significant results for a wide range of spaces of such functions (V. Danchenko, D. Danchenko, Kosukhin, Borodin, Komarov, Chunaev, Novak, Protasov, Kayumov, Kondakova, Shklyaev, Abakumov, Borichev, Fedorovskiy, and others). In the course of these studies it turned out that in other countries some episodic results on approximation by simple partial fractions were proved in the second half of the 20th century (MacLane, Korevaar, Newman, Chui, Elkins and other authors). At the same time, in many papers of these authors the problem of approximation by simple partial fractions was not explicitly stated and the corresponding results were not formulated, but they can be derived from theorems proved in these papers (see Remark 3.2 below). The most striking result was actually obtained by Korevaar [57]; see Theorem 3.1 below: for every bounded simply connected domain $D\subset {\mathbb C}$, any function holomorphic in $D$ can be uniformly approximated with arbitrary accuracy on any compact set in $D$ by simple partial fractions $\operatorname{SF}(\partial D)$ with poles on the boundary $\partial D$ of this domain. Korevaar’s theorem can be considered the starting point for most qualitative results in the theory of approximation by simple partial fractions. A general view of this theory, which also contains non-trivial quantitative results on the rate of approximation, can be obtained from a recent survey [33]. One of the main directions of our research has been determined by our wish to obtain general theorems on the density of quantized approximations in line with Problem 1.1, from which Korevaar’s theorem would follow as a special case. Such theorems are proved in this paper. We note straight away one obvious necessary condition for a positive solution of Problem 1.1. Remark 1.2. For the density of $R(M)$ in $X$, the set $M$ must be all-round: for each non-trivial functional $f\in X^*$ there exists an element $x\in M$ such that $f(x)<0$ ($\operatorname{Re}f(x)<0$ in the case when $X$ is a complex space). Indeed, if $f(x)\geqslant 0$ for each $x\in M$, then also $f(x)\geqslant 0$ for all $x\in R(M)$, and elements $z\in X$ satisfying $f(z)<0$ cannot be approximated by elements of $R(M)$ (geometrically, this means that $R(M)$ lies entirely in one of the half-spaces into which $X$ is divided by the hyperplane $\ker f$). When there is an affirmative answer in Problem 1.1, it is often convenient to divide its proof into two steps: (I) prove that $\overline{R(M)}$ is an additive subgroup of $X$ (that is, every element $-x$, $x\in M$, can be approximated within any accuracy by sums of elements of $M$); (II) in the case when it is already known that $\overline{R(M)}$ is an additive subgroup, prove that this subgroup coincides with $X$. In the finite-dimensional case these two problems are solved in the following way. Theorem 1.3 (see [9]). Let $X$ be a finite-dimensional normed space, and let $M\subset X$. (1) The set $\overline{R(M)}$ is an additive subgroup of $X$ if and only if $M$ is all-round in its linear hull $\operatorname{span}M$. (2) If $M$ is all-round and connected, then $\overline{R(M)}=X$. Of course, the connectedness of $M$ is not a necessary condition in statement (2). This is shown by the example $M=\{-1,\sqrt{2}\,\}$ in the one-dimensional space $X={\mathbb R}$. In [66] it was shown that every infinite-dimensional Banach space $X$ contains a non-compact set $M$, all-round in $\operatorname{span} M$, for which $\overline{R(M)}$ is not an additive subgroup of $X$. In the infinite-dimensional case both statements of Theorem 1.3 fail even for compact curves, as the examples below show. Before we formulate them, we introduce one more notion. The set $\overline{R(M)}$ is a subgroup if each element $-z$, $z\in M$, can be approximated by elements of $R(M)$ within any given accuracy. We distinguish a situation when one can achieve this approximation property for all $z\in M$ simultaneously. An all-round subset $M$ of a Banach space $X$ is said to be minimal if for every neighbourhood $U(x)$ of any element $x \in M$ there is a functional $f\in X^*$ such that $f(y)>0$ ($\operatorname{Re} f(y)>0$ in the complex case) for all $y\in M\setminus U(x)$. Remark 1.4 (see [9]). Let $M$ be an all-round minimal compact subset of $X$. For $\overline{R(M)}$ to be a closed additive subgroup of $X$, it is necessary and sufficient that for every $\varepsilon>0$ one can find $x_1,\dots,x_n\in M$ such that $\|x_1+\dots+x_n\|<\varepsilon$. Now we give examples of connected minimal generating sets $M$ for which $R(M)$ is not dense in the corresponding infinite-dimensional space. Example 1.5 (Borodin [9]). The set $M=\{\alpha_x\colon 0\leqslant x\leqslant 1\}$, where (here and below $I_A$ denotes the indicator function of the set $A$) is an all-round minimal curve in the spaces $L_2[0,1]$ and $L_1[0,1]$, but $\overline{R(M)}$ is not a subgroup of either space. Note that the curve $M$ is rectifiable in $L_1[0,1]$ and non-rectifiable in $L_2[0,1]$. A similar example of an all-round minimal curve that does not generate a subgroup is presented with proof in § 4 below (Example 4.2). Example 1.6. The set $M=\{\pm I_{[0,x]}(t)\colon 0\leqslant x\leqslant 1\}$ is an all-round minimal curve in each space $L_p[0,1]$, $1\leqslant p<\infty$, but $\overline{R(M)}$ coincides with the closed connected additive subgroup $L_p^{\mathbb Z}[0,1]$ consisting of functions from $L_p[0,1]$ taking integer values almost everywhere. In connection with Example 1.6 we note that the structure of a closed additive subgroup of an infinite-dimensional Banach space can be highly non-trivial [4], [36]. For example, there are homotopically non-trivial connected subgroups in a Hilbert space [23], [47]. The subgroup $L_2^{\mathbb Z}[0,1]$ is connected but homotopically trivial, that is, any closed curve in $L_2^{\mathbb Z}[0,1]$ can be contracted to a point inside this subgroup. Problem 1.1 in the general form was first stated and investigated in [9]. The main result of that paper was the following theorem, which actually transfers Theorem 1.3 to the infinite-dimensional case under additional conditions on the set $M$ and the space $X$. Theorem 1.7 (Borodin [9]). Let $X$ be a uniformly convex and uniformly smooth Banach space, and let $\Gamma=\Gamma_1\cup \dots\cup\Gamma_m$ be an all-round minimal subset of $X$ consisting of rectifiable curves $\Gamma_j$. (1) If $m=1$, then $\overline{R(\Gamma)}=X$. (2) If $m>1$, then $\overline{R(\Gamma)}$ is a closed additive subgroup of $X$ containing a vector subspace $L$ of real codimension at most $m-1$, and every functional $f\in L^\perp$ (the annihilator in the realified $X^*$) is constant on each curve $\Gamma_j$, $j=1,\dots,m$. Conditions on $\Gamma$ in this statement were slightly relaxed in [74]. All conditions on $X$ and on $\Gamma_j$ in Theorem 1.7 are essential, as Examples 1.5 and 1.6 show. That case (2) can occur is seen in the example of the set $\Gamma=\{1\}\cup\{-1\}$ in the space $X={\mathbb R}$ (two one-point ’curves’): here $m=2$, $\overline{R(\Gamma)}={\mathbb Z}$, and the subspace $L=\{0\}$ has codimension $m-1=1$. Similar examples in a finite-dimensional Euclidean space can be given for any positive integer $m$. From Theorem 1.7 one can easily derive a weaker version of Korevaar’s theorem for domains $D$ bounded by rectifiable Jordan curves [9]. Namely, we take as $X$ the Hilbert space $\operatorname{AL}_2(\gamma)$, which is obtained by completing the linear space of functions holomorphic in $D$ with respect to the norm $L_2(|dz|)$ on an arbitrary smooth Jordan contour $\gamma\subset D$. It is easy to prove that, under the condition of rectifiability of $\partial D$, the set $\{1/(z-a)\colon a\in \partial D\}$ is an all-round rectifiable curve in $\operatorname{AL}_2(\gamma)$. This curve generates the semigroup $\operatorname{SF}(\partial D)$ of simple partial fractions with poles in $\partial D$. Statement (1) of Theorem 1.7 provides the density of $\operatorname{SF}(\partial D)$ in $\operatorname{AL}_2(\gamma)$, which by virtue of Cauchy’s integral formula implies density in the uniform norm on every compact set inside $\gamma$. The aim of this paper is to review the results related to Problem 1.1 both for arbitrary generating sets $M$ in general Banach spaces $X$ and for specific $M$ in specific function spaces $X$, and also to prove new results. In § 2 Theorem 1.7 is refined and generalized; in particular, the condition of minimality is dropped. In addition, in this section we investigate the possibility of a positive solution to Problem 1.1 for, generally speaking, disconnected generating sets. § 3 contains known and new results on approximation by simple partial fractions in various spaces of functions of a complex variable. Along the way, some well-known theorems (in particular, Korevaar’s theorem) are derived from the new general results in § 2. In § 4 we investigate approximation by a natural generalization of simple partial fractions, namely, sums of shifts of one function. Finally, in § 5, some topics from the theory of approximation by polynomials with integer coefficients are considered, and an attempt is made to outline approaches to the solution of Problem 1.1 in the case of countable generating sets $M$ of general form. 2. General results on the density of a semigroup2.1. Definitions and auxiliary statementsA Banach space $X$ with unit sphere $S(X)$ is said to be uniformly convex if for every $\varepsilon>0$ there exists $\delta>0$ such that
$$
\begin{equation*}
x,y\in S(X), \ \ \biggl\|\frac{x+y}{2}\biggr\|> 1-\delta\quad\Longrightarrow\quad \|x-y\|< \varepsilon.
\end{equation*}
\notag
$$
For any element $x\in X$, the set $J(x)=\{f\in S(X^*)\colon f(x)=\|x\|\}$ is non-empty by a corollary of the Hahn–Banach theorem. A Banach space $X$ is said to be smooth if the set $J(s)$ consists of a single functional $f_s\in S(X^*)$ for every element $s\in S(X)$. A smooth Banach space $X$ is said to be uniformly smooth if for every $\varepsilon>0$ there exists $\delta>0$ such that the inequality holds for all $s\in S(X)$ and $y\in X$, $\|y\|<\delta$. A space $X$ is uniformly smooth if and only if its modulus of smoothness
$$
\begin{equation*}
s(\tau)=\sup\biggl\{\biggl\|\frac{x+y}{2}\biggr\|+ \biggl\|\frac{x-y}{2}\biggr\|-1\colon \|x\|=1, \|y\|=\tau\biggr\}, \qquad \tau\geqslant 0,
\end{equation*}
\notag
$$
has the property $s(\tau)=o(\tau)$ as $\tau\to 0$ (see, for instance, [35], Chap. 3, § 4). The function $s(\tau)$ is convex, positive, increasing and satisfies the inequalities
$$
\begin{equation}
\sqrt{1+\tau^2}-1\leqslant s(\tau)\leqslant \tau, \qquad \tau\in [0,\infty),
\end{equation}
\tag{2.2}
$$
where the right-hand inequality is obvious and the left-hand inequality was proved by Lindenstrauss [61], [35], Chap. 3, § 4, and turns to identity in the case of a Hilbert space. With the help of the modulus of smoothness inequality (2.1) can be generalized and clarified (see, for instance, [76], Chap. 6):
$$
\begin{equation}
\|x+y\|<\|x\|+f_x(y)+2\|x\|\,s\biggl(\frac{\|y\|}{\|x\|}\biggr)
\end{equation}
\tag{2.3}
$$
(here $x\in X\setminus \{0\}$ and $\{f_x\}=J(x)$, $y\in X$). For every Banach space $X$ there is a positive number $\alpha=\alpha(X)$ such that the modulus of smoothness of this space satisfies the inequality
$$
\begin{equation}
s(2\tau)\leqslant \alpha s(\tau), \qquad \tau\in [0,\infty).
\end{equation}
\tag{2.4}
$$
Indeed, in view of the convexity of the modulus of smoothness it is sufficient to establish this inequality near 0, and there it follows from another well-known Lindenstrauss inequality:
$$
\begin{equation*}
\limsup_{\tau\to 0} \frac{s(2\tau)}{s(\tau)}\leqslant 4
\end{equation*}
\notag
$$
(see [61] and [35], Chap. 3, § 4). For $M \subset X$ we set
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cone}M&=\biggl\{\,\sum_{i=1}^n \lambda_i x_i\colon \lambda_i \geqslant 0, \, x_i \in M, \, n \in \mathbb{N} \biggr\} \\ \text{and} \qquad \operatorname{cone}(M,\mathbb{N})&=\biggl\{\,\sum_{i=1}^n\lambda_i x_i\colon \lambda_i \geqslant 0, \, \sum_{i=1}^n \lambda_i \in \mathbb{N}, \, x_i \in M, \, n \in \mathbb{N}\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
In what follows $\operatorname{dist}(x,A):=\inf\{\|x-y\|\colon y\in A\}$ is the distance of the element $x$ to the set $A$. The integer part of a number $a \in \mathbb{R}$ is denoted by $[a]$. All measures are assumed to be non-negative; the full variation of a measure $\mu$ is denoted by $|\mu|$. Lemma 2.1. Let $M$ be an all-round set in a smooth reflexive Banach space $X$. Then $\overline{\operatorname{cone}(M,\mathbb{N})}=X$. Proof. The closed convex subset $\overline{\operatorname{cone}M}$ of the reflexive space $X$ is proximal (see, for instance, [2], Chap. 5), that is, every $x\in X$ has the nearest element $y$ in $\overline{\operatorname{cone}M}$.
If $x\ne y$, then we take a functional $f\in S(X^*)$ that attains its norm at $x-y$ and an element $z\in M$ such that $f(z)>0$ (this is possible because $M$ is all-round). Since the space is smooth, we have
$$
\begin{equation*}
\|x-y-\lambda z\|=\|x-y\|-\lambda f(z)+o(\lambda)\qquad (\lambda\to 0);
\end{equation*}
\notag
$$
hence $\|x-y-\lambda z\|<\|x-y\|$ for small $\lambda>0$. Since $y+\lambda z\in \overline{\operatorname{cone} M}$, this contradicts the fact that $y$ is the nearest element to $x$ in $\overline{\operatorname{cone} M}$.
Thus, $\overline{\operatorname{cone} M}=X$. It remains to prove that every element $x=\sum_{i=1}^n \lambda_i x_i \in \operatorname{cone} M$ can be approximated by elements of $\operatorname{cone}(M,\mathbb{N})$ with an arbitrary accuracy $\varepsilon > 0$. Since $-x_1 \in \overline{\operatorname{cone} M}$, there exist $x_{n+1},\dots,x_{m} \in M$ and $\lambda_{n+1},\dots,\lambda_{m} \geqslant 0$ such that
$$
\begin{equation*}
\biggl\| x_1+\sum_{i=n+1}^m \lambda_i x_i \biggr\| < \varepsilon.
\end{equation*}
\notag
$$
We take $r \in [0,1]$ such that Then and $\|x'-x\| < r\varepsilon \leqslant \varepsilon$. $\Box$ In connection with Lemma 2.1 it is clear that to prove the density of the semigroup $R(M)$ generated by the set $M$, one should learn to approximate an element $x\in\operatorname{cone}(M,\mathbb{N})$ by sums of elements of $M$ itself. The idea of this approximation is in splitting $x=x_1+\dots+x_N$, where each term $x_i$ is a convex combination of elements of a small subset $M_i\subset M$, and in replacing each $x_i\in \operatorname{conv} M_i$ by a nearby element $y_i\in M_i$, so that the sum $y_1+\dots+y_N$ would be close to $x$. In other words, you need to be able to replace the sum of elements of convex hulls of small sets by a close sum of elements of these sets themselves. In uniformly smooth spaces it is possible to do this using ideas due to Troyanski [78]. Lemma 2.2. Let $X$ be a uniformly smooth Banach space with the modulus of smoothness $s(\tau)$. Let $M_i$, $i=1,\dots,N$, be bounded subsets of $X$, and $x_i \in \operatorname{conv}M_i$. We set (1) If $S\leqslant 1$, then there are elements $y_i \in M_i$ such that
$$
\begin{equation}
\biggl\|\,\sum_{i=1}^N(x_i-y_i)\biggr\| \leqslant A \Bigl(\,\max_{1\leqslant i\leqslant N}\operatorname{diam} M_i+ S^{\gamma}\Bigr),
\end{equation}
\tag{2.5}
$$
where $A$ and $\gamma \in (0,1]$ are constants depending only on the function $s(\tau)$. (2) If $S\geqslant 1$, then there are elements $y_i \in M_i$ such that Proof. We take an arbitrary $y_1 \in M_1$. Next we choose $y_{n+1}\in M_{n+1}$, $n=1,\dots, N- 1$, sequentially so that the inequalities hold. Such a choice of $y_{n+1} \in M_{n+1}$ is possible since $x_{n+1} \in \operatorname{conv} M_{n+1}$, that is, $0 \in \operatorname{conv}( x_{n+1} - M_{n+1})$, and hence the set $x_{n+1} - M_{n+1}$ does not lie in any open half-space $\{x \in X\colon f(x) > 0\}$ for any functional $f \in X^* \setminus \{0\}$.
Below, for brevity we denote $\delta=\max_{1\leqslant i\leqslant N}\operatorname{diam} M_i$. Suppose $S\leqslant 1$. The proof of inequality (2.5) for $\sigma_N$ is carried out in accordance to the scheme from [78], Lemma 2. We set $\gamma=\log_\alpha 2$, where $\alpha$ is the number in (2.4). Let $n$ be the maximum number for which $\|\sigma_n\|\leqslant \delta+S^\gamma$. If $n=N$, then (2.5) is true for the constant $A=1$. Othewise, the inequalities
$$
\begin{equation*}
\|\sigma_k\|\leqslant \|\sigma_{k-1}\|+ 2\|\sigma_{k-1}\|\,s\biggl(\frac{\|x_k-y_k\|}{\|\sigma_{k-1}\|}\biggr),\qquad k=n+1,\dots,N,
\end{equation*}
\notag
$$
which are derived from (2.3) in view of (2.7), imply the estimate
$$
\begin{equation*}
\|\sigma_{N}\|\leqslant\|\sigma_n\| \prod_{k=n+1}^N \biggl(1+2s\biggl(\frac{\operatorname{diam} M_k} {\|\sigma_{k-1}\|}\biggr)\biggr)\leqslant (\delta+S^\gamma)\prod_{k=n+1}^N \biggl(1+2s\biggl(\frac{\operatorname{diam} M_k}{S^\gamma}\biggr)\biggr)
\end{equation*}
\notag
$$
(the last inequality is valid since $\|\sigma_{k-1}\|\geqslant \|\sigma_k\|-\|x_k-y_k\|\geqslant \delta+S^\gamma-\delta=S^\gamma$ for $k=n+1,\dots,N$). Transforming the right-hand side of this inequality we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\sigma_{N}\|&\leqslant (\delta+S^\gamma)\exp\biggl(2\sum_{k=n+1}^N s \biggl(\frac{\operatorname{diam} M_k}{S^\gamma}\biggr)\biggr) \\ &\leqslant (\delta+S^\gamma) \exp \biggl(2\sum_{k=n+1}^N s(2^\nu\operatorname{diam} M_k)\biggr)\qquad \biggl(\text{here }\nu=\biggl[\log_2\frac{1}{S^\gamma}\biggr]+1\biggr) \\ &\leqslant (\delta+S^\gamma)\exp\bigl(2\alpha^\nu S\bigr) \leqslant (\delta+S^\gamma)\exp(2\alpha^{\log_2(1/S^\gamma)+1}S) \\ &=(\delta+S^\gamma) \exp\biggl(2\alpha\biggl(\frac{1}{S^\gamma}\biggr)^{\log_2\alpha}S\biggr)= (\delta+S^\gamma)\exp(2\alpha). \end{aligned}
\end{equation*}
\notag
$$
Thus, (2.5) is proved for $\gamma=\log_\alpha 2$, and $A=\exp(2\alpha)$.
Now let $S\geqslant 1$. Let $n$ be the maximum number such that $\sigma_n\leqslant \delta+S$. In the case $n=N$ the inequality (2.6) holds. Otherwise, similarly to the previous arguments we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\sigma_{N}\|&\leqslant\|\sigma_n\|\prod_{k=n+1}^N\biggl(1+ 2s\biggl(\frac{\operatorname{diam} M_k}{\|\sigma_{k-1}\|}\biggr)\biggr) \\ &\leqslant (\delta+S)\prod_{k=n+1}^N \biggl(1+2s\biggl(\frac{\operatorname{diam}M_k}{S}\biggr)\biggr) \\ &\leqslant (\delta+S)\exp\biggl(2\sum_{k=n+1}^N s\biggl(\frac{\operatorname{diam} M_k}{S}\biggr)\biggr) \\ &\leqslant (\delta+S)\exp\biggl(2\sum_{k=n+1}^N s\biggl(\frac{\operatorname{diam} M_k}{2^\nu}\biggr)\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $\nu=[\log_2S]$. Using the convexity of $s$, we continue: As a special case of Lemma 2.2, we obtain the well-known ‘lemma on rounding off the coefficients’, which was proved in the case of uniformly smooth spaces by V. P. Fonf [44] (an equivalent assertion is also contained in the earlier paper by Lindenstrauss [61]). Lemma 2.3 (see [44]). Let $x_1,\dots,x_n $ be elements of a uniformly smooth Banach space $X$ with modulus of smoothness $s(\tau)$, let $x=\lambda_1x_1+\dots+\lambda_Nx_N$, where $\lambda_i\in [0,1]$ ($i=1,\dots,N$), and let $s(\|x_1\|)+\dots+s(\|x_N\|)\leqslant 1$. Then there exist numbers $\theta_i\in \{0,1\}$ such that where the constants $A$ and $\gamma$, $0<\gamma\leqslant 1$, depend only on the function $s(\tau)$. We present one more statement of the type of Lemma 2.2 for Hilbert space. Lemma 2.4. Let $H$ be a Hilbert space, $M_i$, $i=1,\dots,N$, be bounded subsets of $H$, and let $x_i \in \operatorname{conv}M_i$. Then there are elements $y_i \in M_i$ such that Proof. We take arbitrary $y_1 \in M_1$. Next, we choose $y_{n+1} \in M_{n+1}$, $n=1,\dots,N-1$, sequentially so that the inequalities
$$
\begin{equation}
\langle x_{n+1}-y_{n+1},\sigma_n\rangle \leqslant 0, \qquad \sigma_n:= \sum_{i=1}^n(x_i-y_i),
\end{equation}
\tag{2.9}
$$
hold. Such a choice of $y_{n+1} \in M_{n+1}$ is possible because $x_{n+1} \in \operatorname{conv}M_{n+1}$. From (2.9) we obtain and summing these inequalities leads to the desired estimate. $\Box$ In the case when all sets $M_i$ consist of two points, the statement of Lemma 2.4 is well known and, moreover, with the sharp multiplicative constant $1/2$ on the right-hand side of (2.8) [48], Ch. 2, Lemma 1. We do not know whether the constant 1 is sharp in the general inequality (2.8). Without the condition of uniform smoothness, it seems impossible to obtain such general statements on the replacement of a sum of elements of convex hulls of sets by a close sum of elements of these sets themselves. We present a particular statement for arcs of a rectifiable curve in a uniformly convex space. Lemma 2.5 (see [20]). Let $\Gamma$ be a rectifiable curve in a uniformly convex Banach space. For every $\varepsilon>0$ there exists $\delta>0$ such that for any partition of $\Gamma$ into arcs $\gamma_k$ of lengths $|\gamma_k|<\delta$ the inequality holds, where $d(\gamma_k)=\sup\{\operatorname{dist}(x,\gamma_k)\colon x\in \operatorname{conv}\gamma_k\}$. The following lemma allows one to prove the linearity of a closed subgroup in a uniformly smooth space. Lemma 2.6 (convexity of a subgroup; see [9]). Let $G$ be a closed additive subgroup in a uniformly smooth Banach space $X$ with modulus of smoothness $s(\tau)$, $\tau\geqslant 0$. Suppose $a,b\in G$, and for every $\varepsilon>0$ suppose that there are points $x_0,x_1,\dots,x_n\in G$ such that $x_0=a$, $x_n=b$, and $\sum_{k=1}^n s(\|x_k-x_{k-1}\|)<\varepsilon$. Then the whole segment $[a,b]$ lies in $G$. That this statement is accurate for Hilbert space (which has modulus of smoothness $s(\tau)=\sqrt{1+\tau^2}-1=O(\tau^2)$ as $\tau\to 0$) is shown by the subgroup $L_2^{\mathbb Z}[0,1]$ (see Example 1.6 above) consisting of integer-valued functions of $L_2[0,1]$: for any two distinct elements $a,b\in L_2^{\mathbb Z}[0,1]$ the line segment $[a,b]$ does not lie in $L_2^{\mathbb Z}[0,1]$, and at the same time for every $\delta>0$ there are $x_0,\dots,x_n \in L_2^{\mathbb Z}[0,1]$ such that $x_0=a$, $x_n=b$, $\|x_k-x_{k-1}\|<\delta$ for all $k=1,\dots,n$, and the sum $\sum_{k=1}^n \|x_k-x_{k-1}\|^2$ is bounded by a constant depending on $a$ and $b$ only. Lemma 2.6 follows from Lemma 2.3; another proof was given in [9]. 2.2. Semigroup generated by a curveTheorem 2.7 (Borodin [9]). Let $X$ be a uniformly convex Banach space, and let be an all-round minimal subset of $X$ consisting of rectifiable (closed or non-closed) curves $\Gamma_j$. Then $\overline{R(\Gamma)}$ is a subgroup in $X$. The rectifiability of the curves $\Gamma_j$ is essential in this theorem, as Example 1.5 in the space $L_2[0,1]$ shows ($L_2[0,1]$ is uniformly convex). Uniform convexity of the space $X$ is also essential, as Example 1.5 in the space $L_1[0,1]$ shows (in this space, the curve $M$ in Example 1.5 is rectifiable). In the case when $m\geqslant 2$, the subgroup in Theorem 2.7 may not coincide with the whole space, as is shown by the same example $\Gamma=\{1\}\cup\{-1\}$ of two one-point curves in the space $X={\mathbb R}$. In the case $m=1$ the following problem arises. Problem 2.8. Let $\Gamma$ be an all-round (minimal) rectifiable curve in a uniformly convex Banach space $X$. Is it true that $\overline{R(\Gamma)}=X$? If, in addition, $X$ is smooth, then the answer is positive, see Theorem 2.12 below. In this subsection we show that in a uniformly smooth space an all-round rectifiable curve generates a dense semigroup, and no minimality condition is required. Let $X$ be a uniformly smooth Banach space with modulus of smoothness $s(\tau)$, and let $T$ be a compact subset of $[0,1]$. We define a generalized variation of the mapping $F\colon T \to X$ with respect to $s$:
$$
\begin{equation*}
\begin{aligned} \, \operatorname{var}_s(F, \delta) &:= \sup\biggl\{\,\sum_{i=1}^{n-1} s\bigl(\| F(t_{i+1})-F(t_i)\|\bigr)\colon t_1,\dots,t_n \in T, \\ &\qquad\qquad\qquad t_1 \leqslant t_2 \leqslant \cdots \leqslant t_n, \, T \subset \bigcup_{i=1}^n (t_i-\delta, t_i+\delta) \biggr\},\qquad \delta > 0, \\ \operatorname{var}_s(F)&:=\lim_{\delta \to 0+} \, \operatorname{var}_s(F,\delta). \end{aligned}
\end{equation*}
\notag
$$
We also define
$$
\begin{equation*}
\operatorname{br}(F) := \sup\{\| F(t) - F(s)\|\colon t,s \in T,\ (t,s) \cap T=\varnothing\}.
\end{equation*}
\notag
$$
In the case when $T=[0,1]$ this value is zero. Theorem 2.9 (Shklyaev). Let $X$ be a uniformly smooth Banach space with modulus of smoothness $s(\tau)$, $\tau \geqslant 0$, and let $T$ be a compact subset of $[0,1]$. Let $F\colon T \to X$ be a continuous mapping such that $F(T)$ is all-round in $X$. Then where $A$ and $\gamma \in (0,1]$ are constants depending only on the modulus $s(\tau)$. In the case $\operatorname{var}_s(F) \geqslant 1$, the inequality holds with constants $A=55$, $\gamma=1$. Proof. Let $A$ and $\gamma$ be constants from Lemma 2.2. We fix an arbitrary $x\in X$, choose a $\delta$-net $t_0,\dots,t_n \in T$ of $T$, and set $x_\delta:=\sum_{i=0}^n 2F(t_i)$. By Lemma 2.1 there exists an element $x_\varepsilon \in \operatorname{cone}(F(T),\mathbb{N})$ such that Obviously, $x_\delta+x_\varepsilon \in \operatorname{cone}(F(T),\mathbb{N})$. Therefore, there is a measure $\mu$ with finite support in $T$ such that
We can find points $\tau_0 \leqslant \tau_1 \leqslant \cdots \leqslant \tau_{m}$ in $T$ and the corresponding probability measures $\mu_i$ with supports $T_i:=[\tau_{i-1},\tau_i] \cap T$ such that $\mu=\sum_{i=1}^m\mu_i$. It is clear that $\mu(t_j) \geqslant 2$ for $j=0,\dots,n$ by the definition of $x_\delta$, hence each line segment $[\tau_{i-1},\tau_i]$ is contained in some segment $[t_j,t_{j+1}]$, and for every $j$ there is $i$ such that $T_i=\{t_j\}$. We set
$$
\begin{equation*}
x_i=\int_{T} F(t) \, d\mu_i(t) \in \operatorname{conv}F(T_i), \qquad i=1,\dots,m.
\end{equation*}
\notag
$$
We have $x_\delta+x_\varepsilon=\sum_{i=1}^m x_i$. In view of (2.11) and Lemma 2.2 there exist elements $y_i \in F(T_i)$ such that the following estimate holds:
$$
\begin{equation*}
\begin{aligned} \, \operatorname{dist}(x,R(F(T))) &\leqslant \biggl\|x-\sum_{i=1}^m y_i\biggl\| {} \leqslant \biggl\|x-\sum_{i=1}^m x_i\biggr\|+ \biggl\|\,\sum_{i=1}^m x_i-\sum_{i=1}^m y_i\biggr\| \\ &< \varepsilon+A \biggl\{\max_{1 \leqslant i \leqslant m}\, \operatorname{diam}F(T_i)+\biggl(\,\sum_{i=1}^m s \bigl(\operatorname{diam}F(T_i)\bigr)\biggr)^\gamma\biggr\} \\ &=\varepsilon+A \biggl\{\max_{1 \leqslant i \leqslant m}\|F(a_i)-F(b_i)\|+ \biggl(\,\sum_{i=1}^m s\bigl(\|F(a_i)-F(b_i)\|\bigr)\biggr)^\gamma \biggr\}, \end{aligned}
\end{equation*}
\notag
$$
where $a_i,b_i\in T_i$ are such that $\operatorname{diam}F(T_i)=\|F(a_i)-F(b_i)\|$.
For every $i$, the set $T_i$ is contained in some segment $[t_j,t_{j+1}]$. Therefore, every point of $T_i$ differs by at most $\delta$ from either $t_j$ or $t_{j+1}$. Hence either $|a_i-b_i|\leqslant 2\delta$ or there exist $t,s\in T_i$ such that $|a_i-t|\leqslant \delta$, $|b_i-s|\leqslant \delta$, and the interval between $t$ and $s$ does not contain points of $T$. In the first case, $\|F(a_i)-F(b_i)\|\leqslant \omega(F,2\delta)$ (where $\omega$ is the usual modulus of continuity of $F$), and in the second case
$$
\begin{equation*}
\|F(a_i)-F(b_i)\|\leqslant \|F(a_i)-F(t)\|+\|F(t)-F(s)\|+ \|F(s)-F(b_i)\|\leqslant 2\omega(F,\delta)+\operatorname{br}(F),
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\underset{1 \leqslant i \leqslant m}{\max}\|F(a_i)-F(b_i)\|\leqslant 2\omega(F,\delta)+\operatorname{br}(F).
\end{equation*}
\notag
$$
Next, we arrange all the selected points $a_i$, $b_i$: $c_1\leqslant\dots\leqslant c_k$. The set $\{c_i\}$ contains all points $t_j$, hence $\{c_i\}$ is also a $\delta$-net for $T$. Any distinct $a_i$ and $b_i$ are adjacent numbers in the set $\{c_i\}$, and therefore
$$
\begin{equation*}
\sum_{i=1}^m s\bigl(\|F(a_i)-F(b_i)\|\bigr)\leqslant \sum_{i=1}^{k-1} s\bigl(\|F(c_{i+1})-F(c_i)\|\bigr)\leqslant \operatorname{var}_s(F, \delta).
\end{equation*}
\notag
$$
As a result, we obtain
$$
\begin{equation*}
\begin{aligned} \, \operatorname{dist}(x,R( F(T)))&\leqslant \varepsilon+A \bigl(2\omega(F,\delta)+\operatorname{br}(F)+ (\operatorname{var}_s(F,\delta))^\gamma\bigr) \\ &\to A\operatorname{br}(F)+A\bigl(\operatorname{var}_s(F)\bigr)^\gamma \quad\text{as}\ \ \delta,\varepsilon\to 0. \qquad \square \end{aligned}
\end{equation*}
\notag
$$
Remark 2.10. In the case of a Hilbert space $X$, inequality (2.10) can be simplified as Clearly, we cannot prove the density of $R(F(T))$ in $X$ under the assumptions of Theorem 2.9: consider $X={\mathbb R}$ and $F(T)=\{-1,1\}$ for a two-point set $T$. It is unclear, whether one can prove that $\overline{R(F(T))}$ is a subgroup in $X$ under these assumptions (cf. Theorem 2.13 below). In the case $T=[0,1]$, Theorem 2.9 provides density of a subgroup generated by an all-round rectifiable curve in a uniformly smooth space. Here is a more general statement. Theorem 2.11 (Shklyaev). Let $X$ be a uniformly smooth Banach space with modulus of smoothness $s(\tau)$, and let $\Gamma\colon [0,1] \to X$ be an all-round curve. (1) If $\operatorname{var}_s(F)=0$, then $\overline{R(\Gamma)}=X$. (2) In particular, if the modulus of continuity of $\Gamma$
$$
\begin{equation*}
\omega_\Gamma(\tau)=\max \{\|\Gamma(t_1)- \Gamma(t_2)\|\colon |t_1-t_2|\leqslant \tau\}
\end{equation*}
\notag
$$
is such that
$$
\begin{equation*}
s(\omega_\Gamma(\tau))=o(\tau)\quad\textit{as}\ \ \tau\to 0,
\end{equation*}
\notag
$$
then $\overline{R(\Gamma)}=X$. (3) In particular, if $\Gamma$ is rectifiable, then $\overline{R(\Gamma)}=X$. Proof. (1) This statement follows directly from Theorem 2.9.
(2) Let $\varepsilon>0$. Assume $t>0$ is such that $s(\omega_\Gamma(\tau))\leqslant \varepsilon\tau$ for $\tau<t$. It is clear that for any $\delta \in (0,t/2)$ and any points $t_1,\dots,t_n \in [0,1]$ such that
$$
\begin{equation*}
0 \leqslant t_1 \leqslant \cdots \leqslant t_n \leqslant 1, \qquad [0,1] \subset \bigcup_{i=1}^n (t_i - \delta, t_i+\delta),
\end{equation*}
\notag
$$
we have $|t_{i+1}-t_i| < t$, and so $s(\|\Gamma(t_{i+1})-\Gamma(t_i)\|)\leqslant \varepsilon |t_{i+1}-t_{i}|$. Therefore, $\operatorname{var}_s(\Gamma,\delta) \leqslant \varepsilon$. Since $\varepsilon$ can be chosen arbitrarily small, we obtain $\operatorname{var}_s(\Gamma)=0$. Thus, $\Gamma$ satisfies the condition in statement (1), from which it follows that $\overline{R(\Gamma)}=X$.
(3) If the curve $\Gamma$ is parameterized by its natural parameter, then it becomes the Lipschitz image of a line segment, and the condition in (2) is fulfilled due to the uniform smoothness of the space. $\Box$ The curves in Examples 1.5 and 1.6 in $L_2[0,1]$, as well as the curve in Example 4.2 below, satisfy the relation $s(\omega_\Gamma(\tau))=O(\tau)$ as $\tau\to 0$, which shows that statement (2) in Theorem 2.11 is sharp. We present another result on the density of the semigroup generated by a curve in a uniformly convex, but not necessarily uniformly smooth space (see Problem 2.8 above). Theorem 2.12 (Borodin and Shklyaev). Let $\Gamma$ be a rectifiable curve in a uniformly smooth Banach space $X$. Suppose $\Gamma$ has the following property: for every non-zero $x\in X$ there is $y\in \Gamma$ and $\lambda>0$ such that $\|x-\lambda y\|<\|x\|$. (This property holds, for example, if $X$ is smooth and $\Gamma$ is rectifiable.) Then $\overline{R(\Gamma)}=X$. Proof. The density of $\operatorname{cone}(\Gamma,\mathbb{N})$ in $X$ is proved just as in Lemma 2.1. Indeed, uniform smoothness implies the reflexivity of $X$, hence the convex closed cone $\overline{\operatorname{cone}\Gamma}$ is proximal, that is, each $x\in X$ has a nearest point $y\in\overline{\operatorname{cone}\Gamma}$. If $x\ne y$, then by the assumptions of the theorem there exist $t\in [0,1]$ and $\lambda>0$ such that $\|x-y-\lambda\Gamma(t)\|<\|x-y\|$, and this contradicts the fact that $y$ is the nearest point. It remains to prove that every element of $\operatorname{cone}\Gamma$ can be approximated by elements of $\operatorname{cone}(\Gamma,\mathbb{N})$ with arbitrary accuracy, and this is done just as in Lemma 2.1.
Suppose $x\in X$ and $\varepsilon>0$ are fixed. We choose $\delta>0$ corresponding to this $\varepsilon$ by Lemma 2.5. Assume that $\Gamma=\{\Gamma(t)\colon t\in [0,|\Gamma|]\}$ is parameterized by its natural parameter. Just as in the proof of Theorem 2.9, we construct points $0=\tau_0\leqslant\dots\leqslant \tau_m=|\Gamma|$, $\tau_{i+1}-\tau_i<\delta$, and probability measures $\mu_i$ with supports in $[\tau_i,\tau_{i+1}]$ such that the points $x_i=\displaystyle\int\Gamma(t)\,d\mu_i$ satisfy $\Bigl\|x-\displaystyle\sum x_i\Bigr\|<\varepsilon$. Let $\gamma_i=\Gamma([\tau_i,\tau_{i+1}])$. We have $|\gamma_i|<\delta$ and $x_i\in \operatorname{conv}\gamma_i$. We choose the points $y_i\in\gamma_i$ nearest to $x_i$ on the arcs $\gamma_i$. Then we have
$$
\begin{equation*}
\operatorname{dist}(x,R(\Gamma)) \leqslant \Bigl\|x-\sum y_i\Bigr\|\leqslant \Bigl\| x - \sum x_i \Bigr\|+\Bigl\|\,\sum (x_i - y_i)\Bigr\|< \varepsilon+\sum d(\gamma_i)< 2\varepsilon
\end{equation*}
\notag
$$
by Lemma 2.5. $\Box$ The next result generalizes Theorem 1.7 (cf. also Theorem 2.7). Theorem 2.13 (Shklyaev). Let $X$ be a uniformly smooth Banach space with modulus of smoothness $s(\tau)$, and let the curves $\Gamma_i\colon [0,1] \to X$, $i=1,\dots,m$, be such that $\operatorname{var}_s(\Gamma_i)=0$ and the set $\Gamma:=\bigcup_{i=1}^m \Gamma_i([0,1])$ is all-round. Then $\overline{R(\Gamma)}$ is an additive subgroup, and it contains a subspace $L$ of codimension $m-1$ (for $m=1$, we have $\overline{R(\Gamma)}=X$ by Theorem 2.11). In addition, $\operatorname{dist}(x,R(\Gamma)) \leqslant C(\Gamma,s)$ for all $x \in X$, and every functional $f\in L^\perp$ (the annihilator in the realified $X^*$) is constant on each curve $\Gamma_i$, $i=1,\dots,m$. Proof. The existence of the constant $C(\Gamma,s)$ follows from Theorem 2.9: $\Gamma$ can be considered as an image of a set $T$ consisting of several disjoint segments. Now we prove that $\overline{R(\Gamma)}$ is an additive subgroup. We set $a_i=(2i-2)/(2m-1)$, $b_i=(2i-1)/(2m-1)$, $i=1,\dots,m$, and define the curve
$$
\begin{equation}
\Gamma_*(t)= \begin{cases} \Gamma_i\biggl(\dfrac{a_i - t}{b_i - a_i}\biggr), &t \in [a_i, b_i], \\ \Gamma_i(1)+(t - b_i)\dfrac{\Gamma_{i+1}(0) - \Gamma_i(1)}{a_{i+1}-b_i}\,, &t \in (b_i,a_{i+1}), \end{cases}
\end{equation}
\tag{2.12}
$$
where $t\in [0,1]$, $i=1,\dots,m$. Clearly, $\operatorname{var}_s(\Gamma_*)=0$ and $\Gamma_*$ is an all-round set in $X$, hence $\overline{R(\Gamma_*)}=X$ by Theorem 2.9. Consequently, for every $x \in \Gamma$ and every $n\in {\mathbb N}$ there is $y_n \in R(\Gamma_*)$ such that $\| x+y_n\|< 1/n^2$. We set $x_{2i-1} := \Gamma_i(0)$ and $x_{2i}:= \Gamma_i(1)$. From the definition of the curve $\Gamma_*$ it follows that $y_n$ has the form
$$
\begin{equation}
y_n=z_n+\sum_{i=1}^{2m} \lambda_{i} x_i,\qquad z_n \in R(\Gamma), \quad \lambda_i \geqslant 0.
\end{equation}
\tag{2.13}
$$
By Dirichlet’s theorem on joint approximation ([22], Chap. 1, § 5) there are positive integers $l_i$ and $q\leqslant n$ such that
$$
\begin{equation}
|\lambda_i q-l_i| \leqslant \frac{1}{n^{1/(2m)}} \quad \forall \, i=1,\dots,2m.
\end{equation}
\tag{2.14}
$$
Therefore, (2.13) and (2.14) imply that Since $(q-1)x+qz_n+\sum_{i=1}^{2m} l_i x_i \in R(\Gamma)$, the last estimate means that $-x \in \overline{R(\Gamma)}$ for every $x \in \Gamma$, that is, $\overline{R(\Gamma)}$ is an additive subgroup.
Now we prove that $\overline{R(\Gamma)}$ contains a subspace of codimension $m-1$. Since $\operatorname{var}_s(\Gamma_i)=0$, by Lemma 2.6 we have
$$
\begin{equation*}
\overline{R(\pm \Gamma_i)} \supset \{\Gamma_i(s)-\Gamma_i(t)\colon s,t\in [0,1]\},
\end{equation*}
\notag
$$
which implies that
$$
\begin{equation*}
\overline{R(\pm \Gamma_i)} \supset \mathrm{span}_{0}(\Gamma_i) := \biggl\{\,\sum_{k=1}^n \lambda_k u_k\colon u_k \in \Gamma_i, \, \lambda_k \in \mathbb{R}, \,\sum_{k=1}^n \lambda_k=0, \, n \in \mathbb{N}\biggr\}.
\end{equation*}
\notag
$$
Let $L$ be the closure of $\mathrm{span}_{0}(\Gamma_1)+\cdots+\mathrm{span}_{0}(\Gamma_m)$. Clearly, $L$ is a closed subspace of $X$, $L \subset \overline{R(\Gamma)}$. If $L \ne X$, then there is a non-zero functional $f \in L^\perp$. We have $f(x - y)=0$ for any $x, y \in \Gamma_i$, that is, $f$ is constant on each curve $\Gamma_i$. In the case $m=1$ of a single curve the presence of such a functional $f$ contradicts the condition that $\Gamma$ is all-round, so in this case $L=X=\overline{R(\Gamma)}$. Suppose $m > 1$ and $\operatorname{codim}L \geqslant m$, so that $L^\perp$ contains $m$ linearly independent functionals $f_1,\dots,f_m$, where each $f_j$ is constant on each curve $\Gamma_i$. In this case the vectors are either linearly independent, and for some non-trivial linear combination we have or they are linearly dependent, and for some non-trivial linear combination we have In both cases the non-zero functional $\lambda_1f_1+\cdots+\lambda_m f_m$ is constant on the whole of $\Gamma$, which contradicts the condition that $\Gamma$ is all-round. $\Box$ Remark 2.14. For each set $\Gamma=\bigcup_{i=1}^m \Gamma_i$ from the previous theorem, there is an element $x_* \in X$ such that $\overline{R(\Gamma \cup \{x_*\})}=X$. Indeed, for $m=1$ this is obvious, so we assume that $m\geqslant 2$. For the curve $\Gamma_*$ in (2.12) we have
$$
\begin{equation}
R(\Gamma_*)=R(\Gamma)+\operatorname{span}\bigl\{v_i := \Gamma_{i+1}(0) - \Gamma_i(1) \bigr\}_{i=1}^{m-1}.
\end{equation}
\tag{2.15}
$$
We choose a vector $(\alpha_1,\dots,\alpha_{m-1}) \in (0,1)^{m-1}$ such that its sums modulo 1 are dense in the torus $\mathbb{T}^{m-1}=[0,1)^{m-1}$, and we set Let $y=\sum_{i=1}^{m-1} \lambda_i v_i$ be an arbitrary element of $\operatorname{span}\{v_i\}_{i=1}^{m-1}$. For every $\varepsilon > 0$ there are integers $k_1,\dots,k_m$ and a positive integer $k$ such that $|\lambda_i-k\alpha_i-k_i| < \varepsilon$ for all $i=1,\dots,m-1$. It follows that
$$
\begin{equation*}
\biggl\|y-k x_*-\sum_{i=1}^{m-1}k_i v_i\biggr\| \leqslant \varepsilon\sum_{i=1}^{m-1} \| v_i\| \leqslant \varepsilon(m-1)\operatorname{diam}\Gamma \to 0 \qquad (\varepsilon \to 0);
\end{equation*}
\notag
$$
therefore, $y \in \overline{R(\pm\Gamma \cup \{ x_*\})}= \overline{R(\Gamma \cup \{ x_*\})}$ ($\overline{R(\Gamma)}$ is an additive subgroup by Theorem $2.13$). Thus, $\operatorname{span}\{v_i\}_{i=1}^{m-1} \subset \overline{R(\Gamma \cup \{ x_*\})}$, hence $\overline{R(\Gamma \cup \{ x_*\})}=\overline{R(\Gamma_*)}$ by virtue of (2.15). It was proved in Theorem 2.13 that $\overline{R(\Gamma_*)}=X$, hence $\overline{R(\Gamma \cup \{ x_*\})}=X$. 2.3. The semigroup generated by the image of a connected setWe present a result similar to Theorem 2.9 for semigroups generated by images of plane compacta. Compared to Theorem 2.9, here we have to assume more about both the space and the mapping. Theorem 2.15 (Shklyaev). Suppose $X$ is a uniformly smooth Banach space with modulus of smoothness $s(\tau)=O(\tau^2)$ as $\tau\to 0$, $E\subset {\mathbb R}^2$ is a compact set, $f\colon E \to X$ is a Lipshitz mapping with Lipschitz constant $\Lambda$, and suppose $f(E)$ is all-round in $X$. Then for every $x \in X$. Proof. Assume that $E$ lies in the square $[0,1]^2$. Let be the Hilbert curve, which maps the interval $[0,1]$ onto $[0,1]^2$ (see [5]). It was proved in [5] that the square-linear ratio of this curve is equal to $6$, that is,
$$
\begin{equation*}
\sup_{t,s \in [0,1],\ t \ne s}{}\, \frac{|\Gamma_h(t)-\Gamma_h(s)|^2}{|t-s|}=6.
\end{equation*}
\notag
$$
We set $T=\Gamma_h^{-1}(E)\subset [0,1]$. Clearly, $T$ is a compact set by the continuity of $\Gamma_h$. Consider the mapping Since $s(\tau)=O(\tau^2)$, there exists $C(s) > 0$ such that
$$
\begin{equation*}
\begin{aligned} \, s(\| F(t_1) - F(t_2)\|) &\leqslant C(s) \| F(t_1) - F(t_2)\|^2 \\ &\leqslant C(s) \Lambda^2 |\Gamma_h(t_2)-\Gamma_h(t_1)|^2 \leqslant 6\, C(s) \Lambda^2 |t_1 - t_2| \end{aligned}
\end{equation*}
\notag
$$
for all $t_1, t_2 \in T$, from which it follows that $\operatorname{var}_{s}(F) \leqslant 6C(s) \Lambda^2$. Clearly, $\operatorname{br}(F) \leqslant \operatorname{diam}F(T) \leqslant \sqrt{2}\,\Lambda$. Therefore, by Theorem 2.9 we have for every $x\in X$.
The general case of an arbitrary compact set $E\subset {\mathbb R}^2$ reduces to the above case when $E\subset [0,1]^2$ by an affine transformation, as a result of which the constant $\Lambda$ is multiplied by, say, $2\operatorname{diam} E$. Therefore, the final upper estimate depends on $\operatorname{diam} E$ as well. $\Box$ In connection with Theorems 2.11 and 2.15, we arrive at the main problem, which we failed to solve. Problem 2.16. Suppose $X$ is a uniformly smooth Banach space with modulus of smoothness $s$, $s(\tau)=O(\tau^2)$ as $\tau\to 0$, $E$ is a connected compact set in ${\mathbb R}^2$, $f\colon E \to X$ is a Lipschitz mapping, and suppose $f(E)$ is all-round in $X$. Is it true that $\overline{R(f(E))}=X$? If one replaces ${\mathbb R}^2$ by ${\mathbb R}$ in this problem, then the answer is positive, as Theorem 2.11 shows. On the contrary, if one replaces ${\mathbb R}^2$ by ${\mathbb R}^3$, then the answer is negative, as the following example shows. Example 2.17. Let $E$ be a curve of Hausdorff dimension 2 in ${\mathbb R}^3$ such that there is a non-constant function $h\colon E\to {\mathbb R}$ satisfying $|h(u)-h(v)|\leqslant C|u-v|^2$, $u,v\in E$ [19]. We can assume $h(E)=[-1,1]$. Consider the mapping $g\colon [-1,1]\to L_2[0,1]$, $g(x)=\operatorname{sign}x\cdot I_{[0,|x|]}$. This mapping satisfies $\|g(x)-g(y)\|\leqslant C|x-y|^{1/2}$. Consequently, the mapping is Lipschitz, and $f(E)$ is all-round in $L_2[0,1]$. However, $\overline{R(f(E))}$ coincides with the subgroup $L_2^{\mathbb Z}[0,1]$ from Example 1.6 and is not dense in $L_2[0,1]$. In a particular case (sums of shifts of a Lipschitz function on the two-dimensional torus) Problem 2.16 is solved positively in Theorem 4.12 below. Under the additional condition that $\overline{R(f(E))}$ is a subgroup in $X$, in particular, for a symmetric $f(E)$, Problem 2.16 is also solved positively. Theorem 2.18 (Borodin [9]). Suppose $G$ is a closed additive subgroup of a uniformly smooth space $X$, the modulus of smoothness of $X$ satisfies $s(\tau)=O(\tau^2)$ as $\tau\to 0$, $E$ is a connected set in ${\mathbb R}^2$, and $f\colon E\to X$ is a Lipschitz mapping such that $f(E)\subset G$. Then $G$ contains the closed ${\mathbb R}$-linear subspace $L$ spanned by the elements of the form $a-b$, where $a,b\in f(E)$. In particular, if $f(E)$ is all-round, then $G=X$. The proof of Theorem 2.18 is based on Lemma 2.6 and the following assertion. Lemma 2.19 (connectivity feature of plane sets [9]). Let $E\subset {\mathbb R}^2$ be a connected set. Then for any two points $u,v\in E$ and every $\varepsilon>0$, one can find points $z_0=u,z_1,\dots,z_{n-1},z_n=v$ in $E$ such that Connected sets in the three-dimensional space do not have this property [19], while the modulus of smoothness $s$ of a uniformly smooth space cannot have the order of smoothness more than 2 at zero in view of (2.2). Thus, the interaction of Lemmas 2.6 and 2.19 is optimal just for Lipschitz images of connected plane sets in spaces whose modulus of smoothness satisfies $s(\tau)=O(\tau^2)$. In connection with Lemma 2.6 we can consider a more general problem on the density of a semigroup, in which the assumptions on the connectivity features of the generating set are formulated in terms of this set itself, without using the idea of it as an image of some finite-dimensional object. Problem 2.20. Suppose $X$ is a uniformly smooth Banach space with modulus of smoothness $s$, and $M$ is a connected all-round subset of $X$ with the following property: for every pair of points $a,b\in M$ and every $\varepsilon>0$, $M$ contains points $x_0=a,x_1,\dots,x_{n-1},x_n=b$ such that $\sum_{k=1}^n s(\|x_k-x_{k-1}\|)<\varepsilon$. Is it true that $\overline{R(M)}=X$? It is unclear how to solve this problem even in the special case when $X=H$ is a Hilbert space, and the all-round set $M$ is rectifiable, that is, any two points in $M$ can be connected by a curve of length less than some number $L$ in $M$. Nevertheless, it is possible to distinguish a situation in which these problems are solved positively. Theorem 2.21 (Shklyaev). Suppose $X$ and $Y$ are Banach spaces, $A\colon X\to Y$ is a compact operator, and $M\subset X$ is a separable set. If the semigroup $R(M)$ in $X$ has the property where $C$ is an absolute constant, then the closed semigroup $\overline{R(A(M))}$ is a subgroup of $Y$. Proof. Let $\{a_i\}$ be a countable dense subset of $M$. We fix a sequence $\{x_i\}$ of elements of $M$ in which each $a_i$ is repeated infinitely many times. By the hypotheses of the theorem, for every $n$ there is an element $y_n\in R(M)$ such that Thus, the sequence $z_n=\sum_{i=1}^nx_i+y_n\in R(M)$ is bounded, hence the sequence $Az_n$ contains a convergent subsequence. In particular, given $\varepsilon>0$, there is an increasing sequence of numbers $\{n_j\}$ such that for all $j$.
Let $z_{n_1}=g_1+\dots+g_k$, $g_i\in M$. We fix an arbitrary element $x\in M$ and choose numbers $m_0,\dots,m_k$ so that
$$
\begin{equation*}
\|x-a_{m_0}\|_X+\sum_{i=1}^k\|g_i-a_{m_i}\|_X<\varepsilon
\end{equation*}
\notag
$$
(there can be repeating elements among $a_{m_i}$). Then
$$
\begin{equation}
\biggl\|-x-z_{n_1}+\sum_{i=0}^ka_{m_i}\biggr\|_X<\varepsilon.
\end{equation}
\tag{2.17}
$$
Now we choose $n_\nu$ so that Then we have
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|-Ax-\biggl(Az_{n_\nu}-\sum_{i=0}^kAa_{m_i}\biggr)\biggr\|_Y \\ &\qquad\leqslant \biggl\|A\biggl(-x-z_{n_1}+ \sum_{i=0}^ka_{m_i}\biggr)\biggr\|_Y+\biggl\|Az_{n_1}- Az_{n_\nu}\biggr\|_Y<\|A\|\varepsilon+\varepsilon, \end{aligned}
\end{equation*}
\notag
$$
where we used (2.16) and (2.17) in the last inequality.
Thus, $-A(M)\subset \overline{R(A(M))}$. $\Box$ Theorems 2.15, 2.21, and 2.18 entail the following assertion. Corollary 2.22. Suppose $X$ and $Y$ are uniformly smooth Banach spaces whose modules of smoothness decrease as $O(\tau^2)$ as $\tau\to 0$, $E \subset{\mathbb R}^2$ is a connected compact set, $f\colon E\to X$ is a Lipschitz mapping, $A\colon X\to Y$ is a compact operator, and suppose that $A(f(E))$ is all-round in $Y$ (so that $f(E)$ is all-round in $X$). Then $\overline{R(A(f(E)))}=Y$. Below, in § 3.1 we present Korevaar’s theorem as a special case of this corollary. 2.4. The case of a non-connected generating setWhen the generating all-round set $M$ is not connected, it is hardly possible to obtain the density of $R(M)$ in the whole space without additional conditions. However, we can attempt to prove that the closure of $R(M)$ is a subgroup even when $M$ is an all-round compact set of general form. Theorem 2.23 (Borodin [9]). Let $M$ be an all-round minimal compact subset of a Banach space $X$ such that the minimum possible number $n(\varepsilon)$ of elements in its $\varepsilon$-net satisfies for some $\alpha>0$. Then $\overline{R(M)}$ is a subgroup of $X$. This theorem shows that, oddly enough, a subgroup is obtained (that is, the closure of a semigroup is symmetric) for ’slim’ generating compact sets. Apparently, the growth rate of $n(\varepsilon)$ in Theorem 2.23 can be higher (for the compact set $M$ in Example 1.5 in $L_1[0,1]$ we have $n(\varepsilon)=O(1/\varepsilon)$). It is also unclear whether it is possible to do without the condition of minimality. It is essentially used in the proof in accordance with Remark 1.4. If we still wish to obtain the density of a semigroup for, generally speaking, disconnected generating sets, they must be assumed to have elements small in norm and ’sticking out’ in different directions. We say that a subset $M$ of a Banach space $X$ reduces the norm if the inequality $\operatorname{dist}(x,M)<\|x\|$ holds for every non-zero $x\in X$ or, in other words, if every open ball $B(x,\|x\|)$ contains an element of $M$. In the case of reflexive $X$ a norm-reducing set is all-round, since for every non-zero functional $f\in X^*$ the half-space $\{y\colon f(y)> 0\}$ contains the ball $B(x,\|x\|)$, where $x$ is a non-zero element such that $f$ attains its norm on it ($f(x)=\|f\|\cdot \|x\|$; such an element exists by James’s theorem ([35], Chap. 1). The converse is not true in any space: an all-round set may simply not contain elements arbitrarily small in norm, which are always present in a norm-reducing set. The property of a set to reduce the norm was introduced in [14] as a condition necessary for the convergence of various greedy algorithms with respect to this set. Given a norm-reducing set $M\ni 0$ in an arbitrary Banach space $X$, we define the recursive weak semi-greedy algorithm by associating, with every $x=x_0\in X$, the sequence
$$
\begin{equation*}
x_n=x_{n-1}-y_n=x_0-y_1-\dots-y_n, \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
according to the following rules. For each $k$, the element $y_k$ belongs either to $M$ or to $-M$, and for every $y\in M$ and every $n\in {\mathbb N}$ the inequality
$$
\begin{equation*}
|\{k=1,\dots,n\colon y_k=-y\}|\leqslant |\{j=1,\dots,n\colon y_j=y\}|
\end{equation*}
\notag
$$
holds. Then the recursively defined element $x_n$ can be written as where $z_j^{(n)}\in M$. We set $M_n=M\cup \{-z_1^{(n)},\dots,-z_{\nu_n}^{(n)}\}$ and describe finally the step of the algorithm: where $y_{n+1}\in M_n$ and
$$
\begin{equation*}
\biggl\|\frac{x_n}{2}-y_{n+1}\biggr\|\leqslant \frac{\|x_n/2\|+\operatorname{dist}(x_n/2,M_n)}{2}
\end{equation*}
\notag
$$
(such an element $y_{n+1}$ exists since $M_n$ reduces the norm and $0\in M$). If this algorithm converges, that is, $x_n\to 0$, then $x$ can be represented as a series $\sum_{n=1}^\infty y_n$ of elements of $M\cup (-M)$ so that all partial sums of this series are sums of elements of $M$, that is, belong to $R(M)$. Theorem 2.24 (Borodin [14]). Let $M$ be a norm-reducing set in a Hilbert space $H$, and let $0\in M$. Then the recursive weak semi-greedy algorithm converges for every $x\in H$. As a consequence, for every norm-reducing set $M$ in a Hilbert space $H$ the semigroup $R(M)$ is dense in $H$. Theorem 2.24 is one of the few examples of an effective way to find quantized approximations in a fairly general situation. If we do not pursue the convergence of greedy algorithms, then the density of the semigroup generated by a norm-reducing set in the Hilbert space $H$ is not difficult to prove as follows. Suppose $\rho=\operatorname{dist}(x,R(M))>0$ for some $x\in H$. Consider a minimizing sequence $x_n\in R(M)$: $\|x-x_n\|\to \rho$ as $n\to\infty$. We may assume that the sequence $x-x_n$ converges weakly to some $z$. We may also assume that $z\ne0$: otherwise we replace $x$ by $x'=(x+x_k)/2$ for some sufficiently large $k$ (the minimizing sequence $x_n'$ corresponding to this element lies in $B(x',\|x'-x_k\|)\setminus B(x,\rho)$; all points in this set difference are close to $x_k$ for large $k$, hence $x'-x_n'$ are close to $x'-x_k$ and cannot converge weakly to 0). Let $y\in M$ be such that $\|z-y\|^2<\|z\|^2-\varepsilon$, $\varepsilon>0$. Then we have
$$
\begin{equation*}
\begin{aligned} \, \|x-(x_n+y)\|^2&=\|z-y\|^2+\|x-x_n-z\|^2+2\langle z-y, x-x_n-z\rangle \\ &< \|z\|^2-\varepsilon+\|x-x_n-z\|^2+o(1) \\ &=\|z\|^2-\varepsilon+\|x-x_n\|^2+\|z\|^2-2\langle x-x_n, z\rangle+o(1) \\ &=\rho^2-\varepsilon+o(1). \end{aligned}
\end{equation*}
\notag
$$
It turns out that some element of $R(M)$ has distance less than $\rho$ to $x$, which contradicts the definition of $\rho$. The question on the possible generalization of this reasoning to Banach spaces arises. It is easy to show that each non-reflexive space $X$ contains a norm-reducing symmetric set $M$ such that $\overline{R(M)}\ne X$. Indeed, by James’s theorem ([35], Ch. 1) there is a functional $f\in X^*$ not attaining its norm: $|f(x)|\ne\|f\|\cdot \|x\|$ for any non-zero $x\in X$. The kernel $\ker f$ has the property that every $x\notin \ker f$ has no nearest element in $\ker f$, so that $\|x-y\|<\|x-0\|=\|x\|$ for some $y\in \ker f$. Thus, $\ker f$ reduces the norm and is the desired set. We arrive at the following question. Problem 2.25. Is it true that in every norm-reducing set in a reflexive Banach space generates a dense semigroup? One can put the question in another way: does there exist in some Banach space a norm-reducing closed additive semigroup that is not a subspace? Clearly, such a semigroup $G$ should be an antiproximinal set, that is, for every $x$ outside $G$ there is no nearest element in $G$. Indeed, if some $x$ has a nearest element $y\in G$ distinct from $x$, then by the norm-reducing property there would be $z\in G$ such that $\|x-y-z\|<\|x-y\|$, which contradicts the fact that $y$ is the nearest. Thus, Problem 2.25 turns out to be connected with a well-known problem of Konyagin [56]: is it true that there are no closed antiproximinal sets in a reflexive space? There is a fairly wide class of reflexive spaces in which this problem is solved positively. These are the Efimov-Stechkin spaces, that is, Banach spaces $X$ such that for any $x_n\in S(X)$ and $f\in S(X^*)$, the convergence $f(x_n)\to 1$ implies the existence of a fundamental subsequence of $\{x_n\}$ (there are many equivalent definitions of these spaces: [2], Chap 10). Namely, for every closed set $M$ in a Efimov–Stechkin space $X$ the set of points that have a nearest element in $M$ has the second category in $X$ [59]. We have thus proved the following assertion. Theorem 2.26 (Skvortsov). For every norm-reducing set $M$ in a Efimov-Stechkin space $X$, the semigroup $R(M)$ is dense in $X$. Note that for every norm-reducing set $M$ in a reflexive space $X$ the set $\operatorname{cone}M$ of linear combinations of elements of $M$ with positive coefficients is dense in $X$. This is proved in exactly the same way as in Lemma 2.1, and the condition of smoothness of the space is not required precisely just because ’norm-reducing’ is stronger than ’all-round’. We also note that in the case of a symmetric norm-reducing set, a simpler greedy algorithm can be proposed for an effective approximation by a subgroup generated by this set. Given a norm-reducing set $M$ in an arbitrary Banach space $X$, we define the weak semi-greedy algorithm by associating the following sequence with every $x=x_0\in X$: where
$$
\begin{equation*}
y_{n+1}\in M,\qquad \biggl\|\frac{x_n}{2}-y_{n+1}\biggr\|\leqslant \frac{\operatorname{dist}(x_n/2,M)+\|x_n/2\|}{2}
\end{equation*}
\notag
$$
(we select any element $y_{n+1}$ that meets these conditions). Theorem 2.27 (Borodin [14]). Let $M$ be a symmetric norm-reducing set in a Hilbert space $H$, $0\in M$. Then the weak semi-greedy algorithm converges for each $x\in H$, so that $x$ is represented as a series $\sum_{n=1}^\infty y_n$ of elements from $M$. The symmetry condition in this assertion is essential [15]. 3. Approximation by simple partial fractions and their generalizationsFor an arbitrary subset $E$ of the complex plane ${\mathbb C}$, let $\operatorname{SF}(E)$ denote the family of all simple partial fractions with poles in $E$:
$$
\begin{equation*}
\operatorname{SF}(E)=\biggl\{\,\sum_{k=1}^n \frac{1}{z-a_k}\colon a_k\in E, n\in {\mathbb N}\biggr\}.
\end{equation*}
\notag
$$
Our main goal is to present a survey of results on the density of $\operatorname{SF}(E)$ in spaces $X$ of complex functions for various $E$ and $X$. Here are the definitions of the spaces $X$ used below: $A(D)$ is the space of functions holomorphic in the domain $D$, with topology of uniform convergence on compact subsets of $D$; $\operatorname{AC}(K)$ is the space of functions continuous on the compact set $K$ and holomorphic in the interior of $K$, with the uniform norm; $H_p(|z|<1)$ is the Hardy space of functions holomorphic in the unit disc, with the norm
$$
\begin{equation*}
\|f\|_{H_p(|z|<1)}= \lim_{r\to 1}\biggl(\int_0^{2\pi}|f(re^{it})|^p\,dt\biggr)^{1/p};
\end{equation*}
\notag
$$
$H_p(\operatorname{Im} z >0)$ is the Hardy space of functions holomorphic in the upper half- plane, with the norm
$$
\begin{equation*}
\|f\|_{H_p(\operatorname{Im} z>0)}= \sup_{y>0}\biggl(\int_{\mathbb R}|f(x+iy)|^p\,dx\biggr)^{1/p};
\end{equation*}
\notag
$$
$A_p(|z|<1,\omega)$ is the Bergman weight space of functions holomorphic in the unit disc, with the norm
$$
\begin{equation*}
\|f\|_{A_p(|z|<1,\omega)}= \biggl(\iint_{|z|<1}|f(z)|^p\omega(|z|)\, dx\, dy\biggr)^{1/p}.
\end{equation*}
\notag
$$
3.1. Around Korevaar’s theoremTheorem 3.1 (Korevaar [57]). For every bounded simply connected domain $D\subset {\mathbb C}$ the set $\operatorname{SF}(\partial D)$ is dense in $A(D)$. For domains with rectifiable Jordan boundaries this assertion was obtained previously in [63]. The necessity of simple connectedness is quite clear: if, for example, the domain $D$ contains a rectifiable contour surrounding a point $a\in\partial D$, then the function $\theta/(z-a)$ with non-integer $\theta$ cannot be approximated uniformly on this contour by simple partial fractions because of Cauchy’s residue theorem. There are several generalizations of Theorem 3.1 to unbounded domains, which are presented in the next subsection. Remark 3.2. Formally, the papers [57] and [63] do not contain Theorem 3.1. In those papers the following assertion is proved: every function that is holomorphic in a bounded simply connected domain $D$ and has no zeros in $D$ can be approximated with arbitrary accuracy locally uniformly in $D$ by polynomials all of whose zeros lie on the boundary $\partial D$. However, Theorem 3.1 immediately follows from this assertion. Indeed, let $f$ be a function holomorphic in $D$. Then the function where $z_0$ is a fixed point in $D$, has no zeros in $D$ and, according to above assertion, it can be approximated by polynomials $p$ with zeros on $\partial D$ locally uniformly in $D$. Therefore, the function $f=F'/F$ can be approximated by simple partial fractions $p'/p \in \operatorname{SF}(\partial D)$ locally uniformly in $D$. The proof in [57] is very elegant, quite non-trivial, and uses a variety of tools of complex and harmonic analysis. Moreover, simple partial fractions are used in the proof; in particular, the existence of a sequence $r_n\in \operatorname{SF}(\partial D)$ converging to zero in $A(D)$ is established. It is shown that the poles of $r_n$ can be chosen as the images of vertices of some regular polygons inscribed in the unit disc $U$, under a conformal mapping $U\to D$ (this mapping has angular limits almost everywhere on the boundary of $U$ by Fatous theorem), and that these poles are dense on $\partial D$. In our terms, this means that $\overline{\operatorname{SF}(\partial D)}$ is a subgroup in $A(D)$. We present an alternative proof based on Corollary 2.22. Proof of Theorem 3.1. Let $K$ be an arbitrary compact set in $D$. We take two smooth Jordan contours $\gamma$ and $\Gamma$ in $D$ so that (here and below $\operatorname{Int} L$ denotes the inner domain of the Jordan curve $L$) and consider the Hilbert spaces $X=\operatorname{AL}_2(\Gamma)$ and $Y=\operatorname{AL}_2(\gamma)$, each being the completion of the linear space $A(D)$ in the norm $L_2(|dz|)$ on the corresponding contour.
The mapping $f\colon\partial D \to X$, $a\mapsto 1/(z-a)$, is Lipschitz. The operator $A\colon X\to Y$, $Ah=h$, is compact by Cauchy’s integral formula and the compactness principle for holomorphic functions. The image $A(f(\partial D))$ is all-round in $Y$. Indeed, otherwise there is a function $h\in \operatorname{AL}_2(\gamma)$ such that
$$
\begin{equation*}
\operatorname{Re} \int_\gamma\frac{\overline{h(z)}}{z-a}\,|dz|>0, \qquad a\in \partial D.
\end{equation*}
\notag
$$
The left-hand side of this inequality is a function of the variable $a$, harmonic in $\overline{\mathbb C} \setminus \overline{\operatorname{Int}\gamma}$ and equal to zero at infinity. Such a function cannot be positive on $\partial D$ by the extremum principle for harmonic functions.
Thus, Corollary 2.22, as applied to the connected plane compact set $E=\partial D$ and the spaces $X$ and $Y$, mapping $f$ and operator $A$ specified, provides the density of $R(A(f(\partial D)))=\operatorname{SF}(\partial D)$ in $Y=\operatorname{AL}_2(\gamma)$. It turns out that any function from $A(D)$ can be approximated with arbitrary accuracy by fractions from $\operatorname{SF}(\partial D)$ in the norm of $\operatorname{AL}_2(\gamma)$, hence, by Cauchy’s integral formula, also uniformly on $K$. $\Box$ We present some results related directly to Korevaar’s theorem. In 2000, Dolzhenko, who did not not know about Korevaar’s paper, posed the problem on the possibility of uniform approximation on compact sets by simple partial fractions with free poles. The following result similar to Mergelyan’s theorem appeared in the answer. Theorem 3.3 (V. Danchenko and D. Danchenko [32]). Let $K$ be a compact set in ${\mathbb C}$. Then the family $\operatorname{SF}({\mathbb C}\setminus K)$ of simple partial fractions with poles outside $K$ is dense in $\operatorname{AC}(K)$ if and only if ${\mathbb C}\setminus K$ is connected. It is easy to see that sufficiency in Theorem 3.3 follows from Korevaar’s theorem by use of Mergelyan’s theorem (the authors of [32] were also unaware of [57]). Starting from [32], intensive study of approximation by simple partial fractions began in Russia. In particular, the following result was proved, which generalizes both Theorems 3.1 and 3.3. Theorem 3.4 (Borodin [8], [10]). Let $K$ and $E$ be two disjoint compact sets in ${\mathbb C}$, and let $\widehat{E}$ be the union of $E$ with all bounded connected components of the complement ${\mathbb C}\setminus E$. (1) If $K\setminus \widehat{E}$ contains infinitely many points, then $\operatorname{SF}(E)$ is neither all-round nor dense in $\operatorname{AC}(K)$. (2) If $K\subset \widehat{E}$ and ${\mathbb C}\setminus K$ is connected, then $\operatorname{SF}(E)$ is dense in $\operatorname{AC}(K)$. Here assertions (1) and (2) are almost converse to each other. In the intermediate case where $K \setminus \widehat{E}$ is nonempty and finite, $\operatorname{SF}(E)$ may or may not be dense in $\operatorname{AC}(K)$ [8]. The results cited lead naturally to the question of describing universal sets of poles, that is, the sets $E\subset {\mathbb C}$ such that for every compact set $K$ with connected complement $\operatorname{SF}(E\setminus K)$ is dense in $\operatorname{AC}(K)$. For example, Theorem 3.3 means that $E=\mathbb{C}$ is universal. Theorem 3.1 implies universality of the union of closed Jordan contours whose interiors are nested and fill the whole plane. In [8] there is an example of a countable universal set $E$ with unique limit point $\infty$. Clearly, each universal set is unbounded. It seems to be quite difficult to describe all universal sets. However, it is possible to present conditions necessary or sufficient for universality in terms of the set
$$
\begin{equation}
E'(\infty):=\biggl\{\zeta\colon |\zeta|=1,\exists \{a_k\}\subset E, a_k\to \infty\colon \lim_{k\to\infty}\frac{a_k}{|a_k|}=\zeta \biggr\}
\end{equation}
\tag{3.1}
$$
of limit directions at infinity of sequences of points in $E$. Clearly, $E'(\infty)$ is the closed subset of the unit circle. We set Theorem 3.5 (Borodin [16]). Let $E$ be an unbounded subset of the complex plane. (1) If for some positive integer $m$ the convex hull $\operatorname{conv}E_m'(\infty)$ does not contain $0$, then $E$ is not universal; more precisely, $\operatorname{SF}(E\setminus K)$ is not dense in $\operatorname{AC}(K)$ for some disc $K$. (2) If for each positive integer $m$, (here $F^\circ$ denotes the interior of the set $F$), then $E$ is a universal set.(3) In the case where $0\in \operatorname{conv} E_m'(\infty)$ for all positive integers $m$ but condition (3.3) is not satisfied for some $m$, the set $E$ can be universal or not universal alike. The statement (1) in this theorem is proved with the help of the following result. Theorem 3.6 (Borodin and Shklyaev [20]). Let $E\subset \overline{{\mathbb C}}$ be a closed set, let the complement $\overline{{\mathbb C}}\setminus E$ be connected, and for some $R>0$ and $n\in {\mathbb N}$ let the set lie in an angle with vertex 0 of opening smaller than $\pi$. Then for each infinite compact set $K$ disjoint from $E$ and for each $m=1,\dots,n$, the set is not all-round in the space $\operatorname{AC}(K)$, and therefore it is not dense in this space. 3.2. Approximation in unbounded domainsTheorem 3.7 (Elkins [40]). If a simply connected domain $D\subset {\mathbb C}$ lies in a half-plane and does not contain a half- plane, then $\operatorname{SF}(\partial D)$ is dense in $A(D)$. In addition to Korevaar’s machinery, the proof of Theorem 3.7 uses essentially the following subtle result due to Lindwart, Pólya, and Ganelius [45]: if a domain $G$ contains a half-plane and a sequence of polynomials with no zeros in $G$ converges uniformly to a non-zero function on some disc in $G$, then this sequence converges uniformly on compact sets in the whole plane. In this assertion the condition that $G$ contains a half-plane is essential [45], and at present it is not clear how to generalize it to other unbounded domains to obtain analogues of Theorem 3.7 for domains that do not lie in a half-plane (in [40], the role of $G$ was played by one of the complementary domains of $D$ that contains a half-plane). We present an alternative proof of Theorem 3.7 with the help of Corollary 2.22 as well as new results on approximation in unbounded domains by simple partial fractions with poles on the boundaries of these domains. Proof of Theorem 3.7. We assume that $D$ lies in the upper half-plane. The condition ’$D$ does not contain a half-plane’ is equivalent to the condition
Let $K$ be an arbitrary compact set in $D$. We take two smooth Jordan contours $\gamma$ and $\Gamma$ in $D$ suh that
$$
\begin{equation*}
K\subset \operatorname{Int}\gamma\quad\text{and} \quad \gamma\subset \operatorname{Int}\Gamma
\end{equation*}
\notag
$$
and consider the Hilbert spaces $X=\operatorname{AL}_2(\Gamma)$ and $Y=\operatorname{AL}_2(\gamma)$, each being the completion of the linear space $A(D)$ in the norm $L_2(|dz|)$ on the corresponding contour.
Let $z_0$ be some fixed point in $D$, and let $g(z)=1/(z-z_0)$. The set $E=g(\partial D)$ is compact. The mapping
$$
\begin{equation*}
f\colon E \to X,\quad a\mapsto \frac{1}{z-g^{-1}(a)}\,,
\end{equation*}
\notag
$$
is Lipschitz.
The operator $A\colon X\to Y$, $Ah=h$, is compact by Cauchy’s integral formula and the compactness principle for holomorphic functions. Let us show that $A(f(E))$ is all-round in $Y$. Otherwise there is a non-zero function $h\in \operatorname{AL}_2(\gamma)$ such that
$$
\begin{equation*}
\operatorname{Im} \biggl(H(w):= \int_\gamma\frac{\overline{h(z)}}{z-w}\,|dz|\biggr)\geqslant 0, \qquad w\in \partial D.
\end{equation*}
\notag
$$
The function $H$ is holomorphic in $\overline{\mathbb C} \setminus \overline{ \operatorname{Int}\gamma}$ and vanishes at infinity. Its imaginary part is non-negative on $\partial D \cup \{\infty\}$, and therefore also in ${\mathbb C}\setminus D$. We will show that this is impossible. Consider the Laurent expansion for $H$ near $\infty$:
The coefficients $c_k$ cannot all be zero, since
$$
\begin{equation*}
c_k=-\int_\gamma\overline{h(z)}z^{k-1}\,|dz|, \qquad k=1,2,\dots,
\end{equation*}
\notag
$$
and their vanishing means that $h$, as an element of $\operatorname{AL}_2(\gamma)$, is orthogonal to all polynomials. Since polynomials are dense in $\operatorname{AL}_2(\gamma)$, this would mean that $h=0$.
It also follows from the last formula that $|c_k|\leqslant L^k$ for some $L>0$. If $c_1=0$, then $H$ has a zero of at least the second order at infinity, and the image $H(\{\operatorname{Im} z<0\})$ of the lower half-plane, which is a subset of ${\mathbb C}\setminus D$, obviously contains points with negative imaginary part. If the non-zero number $c_1$ is not positive, then the lower half-plane contains a number $\zeta$ such that $\operatorname{Im}(c_1/\zeta)<0$, and for positive $R\to \infty$ we have
$$
\begin{equation*}
\operatorname{Im} H(R\zeta)= \operatorname{Im}\biggl(\frac{c_1/\zeta+o(1)}{R}\biggr)<0.
\end{equation*}
\notag
$$
Now we consider the last case $c_1>0$. Substituting $w=x_n+it_n\notin D$ from (3.4), we obtain
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Im} H(x_n+it_n)&=\frac{-c_1t_n}{x_n^2+t_n^2}+ \operatorname{Im} \sum_{k=2}^\infty \frac{c_k}{(x_n+it_n)^k} \\ &\leqslant \frac{-c_1t_n}{x_n^2+t_n^2}+\frac{|c_2|}{x_n^2+t_n^2}+ \sum_{k=3}^\infty \frac{L^k}{(x_n^2+t_n^2)^{k/2}}<0 \end{aligned}
\end{equation*}
\notag
$$
for all sufficiently large $n$.
So in all cases we arrive at a contradiction. The fact that $A(f(E))$ is all-round in $Y=\operatorname{AL}_2(\gamma)$ is proved. Thus, Corollary 2.22, as applied to the plane set $E$ and the spaces $X$ and $Y$, mapping $f$, and operator $A$ specified, provides the density of $R(A(f(E)))=\operatorname{SF}(\partial D)$ in $Y=\operatorname{AL}_2(\gamma)$. It turns out that every function in $A(D)$ can be approximated with arbitrary accuracy by fractions in $\operatorname{SF}(\partial D)$ in the norm of $\operatorname{AL}_2(\gamma)$, hence, by Cauchy’s integral formula, also uniformly on $K$. $\Box$ We present conditions for an unbounded domain $D$ of a general form which are necessary or sufficient for the density in $A(D)$ of the simple partial fractions with poles on the boundary $\partial D$. These conditions are formulated in terms of powers of the limit directions at infinity (3.1), (3.2) of sequences in the complement of $D$, and almost match each other. Theorem 3.8 (Borodin and Shklyaev). Let $D$ be a simply connected unbounded domain in the complex plane. (1) If for some $m\in {\mathbb N}$ the convex hull $\operatorname{conv}({\mathbb C}\setminus D)_m'(\infty)$ does not contain zero, then $\operatorname{SF}(\partial D)$ is not dense in $A(D)$. (2) If for each positive integer $m$,
$$
\begin{equation}
0\in \bigl(\operatorname{conv}({\mathbb C}\setminus D)_m'(\infty)\bigr)^\circ
\end{equation}
\tag{3.5}
$$
(here $F^\circ$ denotes the interior of the set $F$), then $\operatorname{SF}(\partial D)$ is dense in $A(D)$. (3) In the case where $0\in \operatorname{conv}({\mathbb C}\setminus D)_m'(\infty)$ for all positive integers $m$, but condition (3.5) is not satisfied for some $m$, the set $\operatorname{SF}(\partial D)$ can be dense or not dense in $A(D)$ alike. In [20] this theorem was proved under additional conditions on the boundary of $D$. Proof. (1) By the assumptions of the theorem there is $R>0$ such that the set lies in some angle with vertex 0 of opening less than $\pi$. The set ${\mathbb C}\setminus D$ does not divide the plane, hence by Theorem 3.6 the set $\operatorname{SF}({\mathbb C}\setminus D)$ is not all-round in the space $\operatorname{AC}(K)$ for every infinite compact set $K\subset D$. The set $\operatorname{SF}(\partial D)$ is even less all-round, and therefore it is not dense in $\operatorname{AC}(K)$.
(2) Let us show that the set is all-round in the space $\operatorname{AL}_2(\gamma)$ for every smooth Jordan contour $\gamma\subset D$. Otherwise there is a non-zero function $h\in \operatorname{AL}_2(\gamma)$ such that
$$
\begin{equation*}
\operatorname{Re}\biggl(H(w):= \int_\gamma\frac{\overline{h(z)}}{z-w}\,|dz|\biggr)\geqslant 0, \qquad w\in \partial D.
\end{equation*}
\notag
$$
The function $H$ is holomorphic in $\overline{\mathbb C} \setminus \overline{\operatorname{Int}\gamma}$ and vanishes at infinity. Its real part is non-negative in $\partial D \cup \{\infty\}$, hence it is also non-negative in ${\mathbb C}\setminus D$. Just as above in the proof of Theorem 3.7, it can be shown that $H$ has a non-trivial Laurent series at $\infty$: where $c_m\ne0$. By the assumptions of the theorem, there is $\zeta\in ({\mathbb C}\setminus D)_m'(\infty)$ such that $\operatorname{Re}(c_m/\zeta^m)<0$. Then for the sequence $w_k\in {\mathbb C}\setminus D$, $w_k\to \infty$, satisfying $w_k/|w_k|\to \zeta$ we obtain a contradiction:
$$
\begin{equation*}
\operatorname{Re}H(w_k)=\operatorname{Re}\frac{1}{|w_k|^m} \biggl(\frac{c_m}{(\zeta+o(1))^m}+O\biggl(\frac{1}{|w_k|}\biggr)\biggr)<0, \qquad k\to \infty.
\end{equation*}
\notag
$$
The rest of the proof of the density of $\operatorname{SF}(\partial D)$ in $A(D)$ is carried out in the same way as above, in the proofs of Theorems 3.1 and 3.7, using Corollary 2.22. (3) The domain $D$ from Theorem 3.7 satisfies all assumptions of (3), and $\operatorname{SF}(\partial D)$ is dense in $A(D)$. The upper half-plane $\{\operatorname{Im} z>0\}$ also satisfies all conditions of (3) but $\operatorname{SF}({\mathbb R})$ is not dense in $A(\{\operatorname{Im} z>0\})$: simple partial fractions with poles in the real axis have negative imaginary parts at each point in the upper half-plane. $\Box$ As Theorem 3.8 shows, an important point for the density of $\operatorname{SF}(\partial D)$ in $A(D)$ is that the domain’s complement at infinity, raised to any power, ’sticks out in all directions’ and so is not covered by any half-plane. Meanwhile, the complement to a domain satisfying the assumptions in part (2) of Theorem 3.8 can occupy an arbitrarily small portion of the azimuth at infinity. Corollary 3.9. Let $h_1,h_2\colon{\mathbb R}\to {\mathbb R}$ be continuous functions such that $h_1(x)<h_2(x)$ for all $x$, and let Then for the domain the fractions $\operatorname{SF}(\partial \Pi)$ are dense in $A(\Pi)$. We present one more result (in fact, a corollary of Theorem 3.7), which allows to deal with special domains having symmetries. Theorem 3.10 (Borodin and Shklyaev [20]). Suppose that a simply connected domain $D$ lies in a half-plane, does not contain a half-plane, and $0\in D$. Then for each positive integer $q\geqslant 2$ the set is a simply connected domain and $\operatorname{SF}(\partial\sqrt[q]{D}\,)$ is dense $A(\sqrt[q]{D}\,)$. 3.3. Approximation on unbounded setsSimple partial fractions can be used to approximate functions on unbounded subsets of the complex plane such as lines, rays, and so on. It is quite easy to prove that the semigroup $\operatorname{SF}({\mathbb C}\setminus{\mathbb R})$ of simple partial fractions with poles outside the real line ${\mathbb R}$ is not dense in $L_p({\mathbb R})$ for any $1<p<\infty$. The point is that the set $\{1/(x-a)\colon a\in {\mathbb C}\setminus {\mathbb R}\}$ generating this semigroup is not all-round in view of the inequality
$$
\begin{equation*}
\operatorname{Re} \int_{\mathbb R} \frac{1}{x+i}\, \frac{1}{x-a}\, dx\geqslant 0,\qquad a\in {\mathbb C}\setminus {\mathbb R},
\end{equation*}
\notag
$$
where the function $1/(x+i)$ serves as an element of the conjugate space $L_q({\mathbb R})$, $1/p+1/q=1$. Protasov [70] managed to describe the rather narrow class of functions that can be approximated by simple partial fractions in $L_p({\mathbb R})$. To deal with simple partial fractions on the real line he used a very non-trivial machinery based on the Hilbert transform. Theorem 3.11 (Protasov [70]). Let $1<p<\infty$. Then the closure of the set $\operatorname{SF}({\mathbb C}\setminus {\mathbb R})$ in $L_p({\mathbb R})$ consists precisely of those functions $f\in L_p({\mathbb R})$ that are representable in the form $f=\Phi'/\Phi$, where $\Phi$ is an entire function of order at most $1/q=1-1/p$. At the same time there is a positive result on density for the uniform norm. Theorem 3.12 (Borodin and Kosukhin [18]). For any $u\geqslant 0$ the set $\operatorname{SF}(\{a\colon \left|\operatorname{Im}a\right| >u\})$ is dense in the complex space $C_0({\mathbb R})$ of functions continuous in the real line and tending to zero at infinity, with the uniform norm. The proof of this theorem is essentially based on a powerful result of V. I. Danchenko [29], who actually proved that the closure of $\operatorname{SF}(\{a\colon \left|\operatorname{Im}a\right|>u\})$ in the norm of $C_0({\mathbb R})$ is a subgroup; see Theorem 3.21 below. As Kosukhin showed in [18], for any non-straight angle $\Lambda$ (that is, two rays with a common vertex that do not belong to the same line), the set $\operatorname{SF}({\mathbb C}\setminus \Lambda)$ of simple partial fractions with poles outside $\Lambda$ is not all-round and not dense in $C_0(\Lambda)$. In the case of the semi-axis ${\mathbb R}_+$ density in $L_p$ is obtained for just half the values of $p$. Theorem 3.13 (Borodin [7]). Let $\gamma\in[0,\pi/2)$, $1<p<\infty$. The set is dense in the complex space $L_p({\mathbb R}_+)$ if and only if In particular, Necessity in the second assertion of this theorem is easy to prove: the set $\{1/(x-a)\colon a\in {\mathbb C}\setminus {\mathbb R}_+\}$ is not all-round in $L_p({\mathbb R}_+)$ for $1<p<2$ due to the inequality
$$
\begin{equation*}
\operatorname{Re} \int_{{\mathbb R}_+}\frac{\sqrt{x}}{x+1}\, \frac{1}{x-a}\, dx\geqslant 0,\qquad a\in {\mathbb C}\setminus {\mathbb R}_+,
\end{equation*}
\notag
$$
where the function $\sqrt{x}/(x+1)$ is considered as an element of the dual space $L_q({\mathbb R}_+)$, $1/p+1/q=1$. 3.4. Various function spacesThe question of whether the set $\operatorname{SF}(E)$ of simple partial fractions with poles in a set $E\subset{\mathbb C}$ is dense in some function space with domain of definition $D$ is usually solved in the affirmative by Korevaar’s theorem, provided that $E$ ’surrounds’ $D$ and does not intersect $\partial D$. For example, for any Jordan contour $\gamma$ containing the closed unit disc $\overline U$ inside it, $\operatorname{SF}(\gamma)$ is dense in each Hardy space $H_p(U)$, $1<p<\infty$. Theorem 3.14 (Newman [65]). The simple partial fractions $\operatorname{SF}(C)$ with poles on the unit circle $C$ are not dense in the Bergman space $A_1(|z|<1)$. More precisely, every fraction $r\in \operatorname{SF}(C)$ satisfies Fractions in $\operatorname{SF}(C)$ do not belong to the Bergman spaces $A_p(|z|<1)$ for $p>2$. For $1<p\leqslant 2$ the norms in $A_p(|z|<1)$ of fractions from $\operatorname{SF}(C)$ are also separated from zero by (3.6) and Hölder’s inequality, so that $\operatorname{SF}(C)$ is not dense in those spaces. Fractions in $\operatorname{SF}(C)$ belong to weighted spaces $A_2\bigl(|z|<1,(1-|z|^2)^\alpha\bigr)$ for all $\alpha>0$, and the question on density can be considered. Theorem 3.15 (Abakumov, Borichev, and Fedorovskiy [1]). The simple partial fractions $\operatorname{SF}(C)$ with poles on the unit circle $C$ are dense in the space $A_2\bigl(|z|< 1, (1-|z|^2)^\alpha\bigr)$ if and only if $\alpha>1$. Theorem 3.16 (Chui [25]). Let $D$ be a Jordan domain with a rectifiable boundary $\Gamma$. For $q>2$, the simple partial fractions $\operatorname{SF}(\Gamma)$ with poles in $\Gamma$ are dense in the Bers space $B_q(D)$ of functions holomorphic in $D$, with the norm where $\lambda_D(z)$ is the Poincaré metric on $D$, which in this case plays the role of a weight function and can be expressed by the formula where $g$ is an arbitrary conformal mapping of $D$ onto the unit disc. Since $B_2(|z|<1)=A_1(|z|<1)$, the sharpness of the condition $q>2$ in Theorem 3.16 is confirmed by Theorem 3.14. In 2014 Nasyrov posed the following problem: are the simple partial fractions with poles on the unit circle dense in the complex space $L_2[-1,1]$? This problem was fixed in [10] and was solved in the negative by Komarov [51], who considerably generalized and clarified his result subsequently (see [55]). Theorem 3.17 (Komarov [51], [55]). The simple partial fractions $\operatorname{SF}(C)$ with poles on the unit circle $C$ are not dense in the complex space $L_p[-1,1]$ for $p\geqslant 1$. More precisely, every fraction $r\in\operatorname{SF}(C)$ of degree $n$ satisfies for any $p\geqslant 1$. The power $1-1/p$ in (3.8) is sharp, as the fraction shows. Moreover, the inequality (3.8) for $p=1$ entails Theorem 3.14, however, with a slightly smaller constant on the right-hand side of (3.6).The problem arises about the density of the set $\operatorname{SF}(C)$ in the weight spaces $L_p([-1,1],(1-x^2)^\alpha)$. Ershov proved density for $p=2$ and $\alpha>1$ in 2022. It is interesting to note that $\alpha>1$ are exactly those values of $\alpha$ for which $\{1/(x- a)\colon a\in C\}$ is bounded in $L_2([-1,1],(1-x^2)^\alpha)$. Simple partial fractions $\operatorname{SF}(\{a\colon \operatorname{Im}a<0\})$ with poles in the lower half-plane are not dense in the Hardy spaces $H_p(\{\operatorname{Im}z >0\})$ in the upper half-plane, since their values have negative imaginary part at all points of the upper half-plane. However, derivatives of simple partial fractions are devoid of this shortcoming. Theorem 3.18 (Dyuzhina [38]). The derivatives of simple partial fractions are dense in all Hardy spaces $H_p$ in the upper half-plane for $1<p<\infty$, and also in the space of functions holomorphic in the upper half-plane, continuous on its closure, and vanishing at infinity. It is unclear whether this statement is true for $p=1$. 3.5. The rate of approximation and extremal problemsAfter the above qualitative results, it is quite natural to consider the problem of estimating the rate of approximation by simple partial fractions, depending on the degree of the approximating fraction. Such estimates and various constructive methods of approximation were rather explicitly presented in the recent survey [33]. Without going into this topic, we present two results typical for it. Let
$$
\begin{equation*}
\rho_n(f):=\min\{\|f-r\|_{C[-1,1]}\colon r\in \operatorname{SF}({\mathbb C}\setminus [-1,1]),\deg r\leqslant n\}
\end{equation*}
\notag
$$
be the least uniform deviation of the function $f\in C[-1,1]$ from the simple partial fractions of degree at most $n$. Theorem 3.19 (Kosukhin [58]). For every $f\in C[-1,1]$ and every $n=1,2,\dots$, where $C(t)=12(1+t)e^{2t}$. This theorem, when compared with classical Jackson’s theorems, shows that simple partial fractions approximate in $C[-1,1]$ in about the same way as polynomials. However, there are also special topics, such as the approximation of constants. Theorem 3.20 (Komarov [49]). For every $\lambda>0$, and this rate of approximation is attained on the simple partial fractions that interpolate the constant function $\lambda$ at Chebyshev nodes in $[-1,1]$. It should be noted that approximation of constants is quite meaningful from the physical point of view: we approximate the electrostatic field of constant intensity by a field created by $n$ identical charges. In the constructive theory of approximation by simple partial fractions, there are interesting extremal problems related to the rate of approximation of the function zero in various norms. Let $E$ be a subset of the complex plane, $a$ be a fixed point in $E$, $\operatorname{SF}_n(E)$ denote the set of simple partial fractions of degree at most $n$ with all poles in $E$ and $X$ be some Banach space of functions containing $\operatorname{SF}(E)$. We set
$$
\begin{equation*}
\rho_n(a,E,X)=\inf\{\|r_n\|_{X}\colon r_n\in \operatorname{SF}_n(E), r_n(a)=\infty\}.
\end{equation*}
\notag
$$
In fact, $\rho_n(a,E,X)$ is the distance from the function $-1/(z-a)$ to $\operatorname{SF}_{n-1}(E)$ in the space $X$. Convergence of $\rho_n(a,E,X)$ to zero as $n\to \infty$ (in this case, the poles of fractions realizing the values of $\rho_n$ form ’asymptotically neutral families of points’ from $E$ in Korevaar’s terminology) means that $-1/(z-a)$ belongs to the closure of $\operatorname{SF}(E)$ in $X$, and often plays a key role in the proof of density of $\operatorname{SF}(E)$ in $X$. Estimates of this quantity for various concrete pairs of $E$ and $X$ have been obtained by many mathematicians. Here are the most striking results. Theorem 3.21 (Danchenko [29]). For an arbitrary point $a\in {\mathbb C}\setminus {\mathbb R}$, This is a very difficult and beautiful theorem. Formally, [29] was devoted to the solution of another problem with a rich history, posed by Gorin, on estimating the minimum possible distance to the real axis ${\mathbb R}$ of a pole of a simple partial fraction $r_n$ of degree $n$ under the condition $\|r_n\|_{C({\mathbb R})}\leqslant 1$. It is easy to show that this problem is equivalent to the problem of estimates for $\rho_n(i,{\mathbb C}\setminus {\mathbb R},C({\mathbb R}))$. For any $p\in (1,\infty)$, $n\in {\mathbb N}$, and $a\in {\mathbb C}\setminus {\mathbb R}$ the results in [6] imply that
$$
\begin{equation*}
\frac{2^{2/p}\pi^{1/p}}{p\left|\operatorname{Im}a\right|^{1/q}}\, \mathrm{B}\biggl(\frac{q}{2}\,,\frac{q}{2}\biggr)^{1/q}\leqslant \rho_n(a,{\mathbb C}\setminus {\mathbb R},L_p({\mathbb R}))\leqslant \biggl(\frac{\pi q\,2^{2-p}}{p\,\mathrm{B}(p/2,p/2)}\biggr)^{1/p} \frac{1}{\left|\operatorname{Im}a\right|^{1/q}}\,,
\end{equation*}
\notag
$$
where $1/p+1/q=1$ and $\mathrm{B}(\alpha,\beta)= \displaystyle\int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt$ is Euler’s beta-function. Here we have no convergence to zero as $n\to\infty$, which corresponds to the fact noted above that $\operatorname{SF}({\mathbb C}\setminus {\mathbb R})$ is not dense in $L_p({\mathbb R})$. Earlier weaker estimates and non-decreasing of the values $\rho_n(a,{\mathbb C}\setminus {\mathbb R},L_p({\mathbb R}))$ to zero were established by Danchenko [29]. For $p=2$, the above upper and lower estimates coincide:
$$
\begin{equation*}
\rho_n(a,{\mathbb C}\setminus {\mathbb R},L_2({\mathbb R}))= \biggl(\frac{\pi}{\left|\operatorname{Im}a\right|}\biggr)^{1/2}= \biggl\|\frac{1}{x-a}\biggr\|_{L_2({\mathbb R})}.
\end{equation*}
\notag
$$
For an arbitrary point $a\in {\mathbb C}\setminus {\mathbb R}_+$ outside the semi-axis ${\mathbb R}_+=[0,+\infty)$ and for all sufficiently large $n$, the following estimates hold (see [7]):
$$
\begin{equation*}
\frac{C_1(a)}{\sqrt{\ln n}}\leqslant \rho_n(a,{\mathbb C}\setminus {\mathbb R}_+,L_2({\mathbb R}_+)) \leqslant \frac{C_2(a)\ln\ln n}{\sqrt{\ln n}}\,.
\end{equation*}
\notag
$$
Theorem 3.21 is used in the proof of the upper estimate. Using the construction proposed in [30], Chunaev [27] ascertained the asymptotic behaviour of $\rho_n(a,{\mathbb C}\setminus[-1,1],C[-1,1])$ (for real $a$ lying not too close to the interval $[-1,1]$ and for the narrower class of real simple partial fractions), his result was supplemented by Komarov [53]. For other results of this type, see the survey [33]. In the light of Korevaar’s theorem and its generalizations (see §§ 3.1 and 3.2) it seems reasonable to consider the problem of estimating the quantities $\rho_n(a,\partial D, \operatorname{AC}(K))$ for various simply connected domains $D$, compact sets $K\subset D$, and points $a\in \partial D$. For example, in the case of the strip $D=\Pi:=\{z\colon\left|\operatorname{Im}z\right|<1\}$ (we can apply, say, Corollary 3.9 to this domain) there is an upper estimate
$$
\begin{equation*}
\rho_n(a,\partial\Pi,\operatorname{AC}(K)) \leqslant L(K)\,\frac{\ln^2n}{n}
\end{equation*}
\notag
$$
for every compact set $K\subset \Pi$, every point $a\in\partial \Pi$ and all $n>1$ (see [20]). Dyuzhina [38] proved that for all $1< p \leqslant \infty$, sufficiently large $n$, and each $a$ in the lower half-plane $\Pi_-$ there is an estimate
$$
\begin{equation*}
\inf\{\|r_n'\|_{H_p(\Pi_+)}\colon r_n\in \operatorname{SF}_n(\Pi_-), r_n(a)=\infty\}\leqslant \frac{128}{\left|\operatorname{Im}a\right|^{2-1/p}}\, \frac{(\ln n)^{2-1/p}}{n^{1-1/p}}\,.
\end{equation*}
\notag
$$
For $p=\infty$, this estimate follows from V. Danchenko’s results and is sharp in order as $n\to\infty$ [29]. In the special case when $E$ is the unit circle, the problem of the uniform distribution of the poles of the extremal simple partial fraction is often posed:
$$
\begin{equation}
\rho_n(1,\{w\colon |w|=1\},X)=\biggl\|\frac{nz^{n-1}}{z^n-1}\biggr\|_X?
\end{equation}
\tag{3.9}
$$
For the Bergman space $X=A_1(|z|<1)$ this is the well-known and still unsolved Chui conjecture [24]. In [1] the equality (3.9) was proved for the weighted Bergman space $X=A_2\bigl(|z|< 1, (1-|z|^2)^\alpha\bigr)$ for $0<\alpha\leqslant 1$. In the case when $X=L_p(|z|= r)$ for $0<r<1$ and $1\leqslant p\leqslant \infty$, the question (3.9) was originally posed in [72] but, in fact, in a weaker setting. For $p=\infty$ the answer is positive [71]. For an arbitrary $p$ some advance was made in [73]. 3.6. Generalizations of simple partial fractionsAs already noted, approximations by simple partial fractions have a natural physical interpretation: the intensity of an arbitrary plane electrostatic field is approximated by the intensity of a field produced by equal charges of the same sign. It is natural to consider the problem on such an approximation for electrostatic fields in spaces of higher dimensions. This is how the analogue of Korevaar’s theorem looks like in the three-dimensional case. Theorem 3.22 (Piele [69]). Let $D$ be a bounded domain with connected complement in ${\mathbb R}^3$ such that its boundary contains asymptotically neutral families of points, that is, families $\{y^n_1,\dots,y^n_n\}_{n\in {\mathbb N}}\subset \partial D$ such that uniformly on compact subsets of $D$ for some real constants $C_n$. Then for every harmonic function $f$ in $D$ there are families of points $\{x^n_1,\dots,x^n_n\}_{n\in {\mathbb N}}\subset \partial D$ depending on $f$ and constants $C_n(f)$ such that uniformly on compact subsets of $D$. Note that here we are dealing not with approximating the intensity, but with approximating (up to constants) the potential of an arbitrary electrostatic field by the potentials of the fields created by a finite number of identical point charges positioned at the points $x^n_k$. In other words, not the sums of $1/|x-x^n_k|$ but their gradients are analogues of simple partial fractions here. From the general point of view, Theorem 3.22 has the type of assertion on the density of a subgroup: it is easy to show (and this is really done in [69]) that asymptotically neutral families of points are dense in $\partial D$, hence every function $-1/|x-a|$ can locally uniformly be approximated in $D$ up to constants by sums It was shown in [68] that asymptotically neutral families exist for any domain whose boundary is a Lyapunov surface, admitting at each point a local parametrization for which all partial derivatives of the third order satisfy Hölder’s condition. If it were possible to prove that every bounded domain with connected complement in ${\mathbb R}^3$ admits asymptotically neutral families of points, then Theorem 3.22 would turn to a complete analogue of Korevaar’s theorem. Another generalization of simple partial fractions is the so-called $h$-sums. V. Danchenko [31] initiated the study of the approximation properties of the sums where $h$ is a fixed function of the complex variable, holomorphic in a neighborhood of the origin, and $\lambda_1,\dots,\lambda_n$ are arbitrary complex numbers. These sums are a natural generalization of simple partial fractions (which correspond to the special case $h(z)=1/(z-1)$). Later on, the interpolation properties and other properties of the sums (3.10) were investigated by V. Danchenko and his students [26], [28], [33], [52].In particular, the following result was proved in [31]. Theorem 3.23 (V. Danchenko [31]). Let $h$ be a function holomorphic in the unit disc $U=\{z\colon |z|<1\}$, $h(z)=\sum_{n=0}^\infty h_n z^n$. Suppose the function $f(z)=\sum_{n=0}^\infty f_n z^n$ satisfies $|f_n|\leqslant |h_n| a^n$, $n=0,1,2,\dots$, for some $a>0$ (so that $f$ is holomorphic in the disc $|z|<1/a$). Then there are $\lambda_{nk}$, $k=1,\dots,n$, $n=1,2,\dots$, such that $|\lambda_{nk}|\leqslant 2a$ for all $n$ and $k$, and the sums converge to $f(z)$ locqlly uniformly in the disc $|z|<1/(2a)$. At the same time Danchenko invented a method for calculating the parameters $\lambda_{nk}$ (in fact, $H_n$ interpolates $f$ at the origin with multiplicity $n$) and estimated the rate of convergence of $H_n$ to $f$ on compact subsets of $|z|<1/(2a)$. In connection with Theorem 3.23, a natural question arises on the possibility of approximating all functions holomorphic in the same disc as $h$ by sums (3.10) generated by $h$. Given a holomorphic function $h$ in $U$, the function $\lambda h(\lambda z)$ is holomorhic in $U$ for any $\lambda\in \overline{U}$. Therefore, it is natural to set the problem of density in $A(U)$ for the set
$$
\begin{equation*}
S(h,E)=\biggl\{\,\sum_{k=1}^n \lambda_k h(\lambda_k z)\colon \lambda_k\in E, n\in {\mathbb N}\biggr\},
\end{equation*}
\notag
$$
where $h$ is holomorphic in $U$ and $E$ is a compact subset of the closed unit disc $\overline{U}$. In what follows, $\widehat{E}$ denotes as usual the union of a compact set $E$ with all bounded connected components of the complement ${\mathbb C} \setminus E$. Theorem 3.24 (Borodin [13]). Let $E\subset \overline{U}$ be a compact set. (1) If $0\notin \widehat{E}$, then the sums $S(h,E)$ are not dense in $A(U)$ for any function $h\in A(U)$. (2) If $0\in \widehat{E}\setminus E$ or $0\in E^\circ$, then $S(h,E)$ is dense in $A(U)$ for every function $h\in A(U)$ all of whose Taylor coefficients are distinct from zero. (3) In the case when $0\in E\setminus E^\circ$ the set $S(h,E)$ can be dense or not dense in $A(U)$ alike. In the case of $h(z)=1/(z-1)$, this theorem leads to the following assertion: the simple partial fractions $\sum_{k=1}^n\dfrac{1}{z-a_k}$ with poles $a_k\in E^{-1}=\{\lambda^{-1}\colon \lambda\in E\}\subset {\mathbb C}\setminus U$ are dense in $A(U)$ under the condition $U\subset \widehat{E^{-1}}\setminus E^{-1}$. This is a corollary of Korevaar’s theorem. It would be interesting to clarify statement (3) of Theorem 3.24 in terms of the limit values at zero of the arguments of points in $E$, as in Theorem 3.8. Here is another result on approximation in the closed disc. Theorem 3.25 (Borodin [11]). Let $h$ be a function holomorphic in the unit disc $U$ with Taylor expansion $\sum_{n=0}^\infty h_n z^n$ such that Then the sums $S(h,\partial U)$ are dense in $\operatorname{AC}(\overline{U})$. The conditions $h_n\ne0$ in Theorems 3.24 and 3.25 are necessary: if $h_n=0$ for some $n$, then each sum (3.10) has $n$th Taylor coefficient equal to zero, so that the function $z^n$ cannot be approximated by these sums. In the context of the theory of approximation by simple partial fractions and their generalizations it seems quite legitimate to consider the problem of approximation by differences $r_1- r_2$ of simple partial fractions (logarithmic derivatives of rational functions) under the condition that the poles of a fraction $r_1$ lie in a prescribed set $E^+$ and the poles of $r_2$ lie in another set $E^-$, where $E^+\cap E^-=\varnothing$. This setting has a natural physical interpretation. If we recall that the simple partial fraction with poles $\{a_k\}$ is the complex conjugate to the intensity of the plane electrostatic field produced by equal charges of the same sign positioned at the points $a_k$ ([60], Chap. 3, § 2), then the above differences $r_1-r_2$ correspond to the fields created by equal charges of different signs positioned in a natural way in two different regions in the complex plane (’capacitor’). We present a result on uniform approximation on a compact set by fractions $r_1-r_2$ of the above type. It is remarkable that for the density of such fractions in $\operatorname{AC}(K)$ it is not necessary that $E^+$ and $E^-$ ’encircle’ the compact set $K$. Theorem 3.26 (Borodin [10]). Let $E^+,E^-\subset{\mathbb C}$ be the connected components of the boundary of a doubly connected domain $D\subset \overline{\mathbb C}$ and let $K\subset {\mathbb C}\setminus \overline{D}$ be a compact set with connected complement. Then the fractions of the form are dense in $\operatorname{AC}(K)$. Corollary 3.27. Let $E^+$ and $E^-$ be mutually exterior, closed Jordan curves and let the compact set $K$ with connected complement lie in the union of the interiors $\operatorname{Int} E^+\cup \operatorname{Int}E^-$ of these curves. Then the functions of the form are dense in $\operatorname{AC}(K)$. It is quite easy to give an example showing that the condition on the compact set $K$ is essential in this Corollary. Let $E^+$ and $E^-$ be two mutually exterior circles with centres $-1$ and $1$, respectively, and let $K=\{0\}$. Then all fractions in Corollary 3.27 have positive real parts at $0$ and hence are not dense in $\operatorname{AC}(K)={\mathbb C}$. Note that the problem of the density of the above differences $r_1-r_2$ of simple partial fractions is solved much easier under the assumption $E^+\cap E^-=E\ne\varnothing$, since in this case we have an entire subgroup generated by the set $\{\pm 1/(z-a)\colon a\in E\}$, and Theorem 2.18 can be used. The first theorems of this kind for special sets $E$ were obtained in [64]. We present a fairly general result on the approximation by plus-minus shifts of one holomorphic function with singularities. Theorem 3.28 (Borodin [9]). Let $K\subset {\mathbb C}$ be a compact set with connected complement, $f(z)$ be a non-zero function holomorphic outside another compact set $F$ such that $f(\infty)=0$, and $E$ be a compact connected subset of the unbounded connected component of the set $\{a\in {\mathbb C}\colon (F+a)\cap K=\varnothing\}$. If $E$ is a uniqueness set for harmonic functions (any two harmonic functions in a domain $D\supset E$ that coincide on $E$, coincide everywhere in $D$), then the finite sums of the functions $\pm f(z-a)$, $a\in E$, are dense in $\operatorname{AC}(K)$. In the special case $f(z)=1/z$ we obtain the following assertion: if a compact set $K$ has a connected complement and a connected compact set $E\subset {\mathbb C}\setminus K$ is a uniqueness set for harmonic functions, then the finite sums of the functions $\pm 1/(z-a)$, $a\in E$, are dense in $\operatorname{AC}(K)$. Note also that approximation by such sums on compact sets $K$ with the natural constraint $E\subset{\mathbb C}\setminus K$ on their poles are interesting from the quantitative point of view: the rate of approximation in this case is much better than when approximating by simple partial fractions [50]. Theorem 3.28 leads to the idea of the most natural generalization of simple partial fractions – sums of shifts of one function. The problem of the density of such sums is considered in the next section. 4. Approximation by sums of shifts of one functionThe completeness (density of the linear span) of the system of shifts of one function or one vector in a sequence space has been under investigation since the classical paper by Wiener [79]. A necessary and often sufficient condition for completeness is that the ‘coefficients’ of the function in its ‘expansion’ in the ‘eigenfunctions’ of the shift operator are distinct from zero. For example, in the case of the unit circle these are the usual trigonometric Fourier coefficients, and in the case of the real line, these are the values of the Fourier transform. For the density of the sums of shifts with coefficients 1 or $\pm 1$, this condition is also necessary. 4.1. Shifts on the circleFor any $2\pi$-periodic real-valued function $f\in L_p({\mathbb T}:=[0,2\pi))$ the sums of shifts
$$
\begin{equation}
\sum_{k=1}^N f(t+a_k), \qquad a_k\in \mathbb{R}, \quad N=1,2,\dots,
\end{equation}
\tag{4.1}
$$
cannot be dense in the whole space $L_p({\mathbb T})$: if the mean value $\displaystyle\int_{\mathbb T} f(t)\, dt$ of $f$ is equal to $\alpha$, then sums (4.1) cannot approximate functions whose mean value is not in the set $\{n\alpha\colon n\in{\mathbb N}\}$. Therefore, we can only raise the question of density in the subspace $L_p^0({\mathbb T})$ of functions with zero mean value. The semigroup (4.1) is generated by the closed curve $\{f(t+a)\colon a\in [0,2\pi]\}$, so we can apply Theorem 2.11. This curve is all-round in the real space $L_p^0({\mathbb T})$ if and only if the Fourier coefficients $c_n$ of $f$ are non-zero for $n\in {\mathbb Z}\setminus \{0\}$. The rectifiability of this curve in, say, $L_2^0({\mathbb T})$ is equivalent to $\sum_{n\in \mathbb{Z}}|n|^2 |c_n|^2<\infty$. Thus, Theorem 2.11 provides density under the conditions indicated. However, a more accurate result can be proved. Theorem 4.1 (Borodin [11]). Let $1\leqslant p<\infty$, and let $f$ be a $2\pi$-periodic function in the real space $L_p({\mathbb T})$ with Fourier expansion such that (a) $c_0=0$ and $c_n\ne 0$ for $n\in {\mathbb Z}\setminus \{0\}$; (b) $\sum_{n\in \mathbb{Z}}|n|\, |c_n|^2<\infty$ for $1\leqslant p\leqslant 2$ or $\sum_{n\in \mathbb{Z}}|n|\,|c_n|^q<\infty$ for $2\leqslant p <\infty$ ($1/p+1/q=1$). Then the sums (4.1) are dense in the real space The difference of indicators $f=I_{[0,\alpha]}-I_{[2\pi-\alpha,2\pi)}$ satisfies condition (a) in Theorem 4.1 for almost all $\alpha$, but sums of shifts (4.1) take only integer values and are not dense in $L_p^0(\mathbb{T})$. This example shows that condition (b) in Theorem 4.1 cannot be replaced by $|c_n|=O(1/n)$ ($n\to \infty$). A linear function provides another supporting example. Example 4.2 (Borodin [11]). Consider the function $f\colon {\mathbb T}\to {\mathbb R}$, $f(t)=t-\pi$, extended periodically to the whole real line. It has the Fourier expansion Moreover, it was shown in [11] that for any function $f$ from $L_p^0({\mathbb T})$ ($2\leqslant p<\infty$), whose Fourier coefficients satisfy $|c_n|\geqslant 1/|n|$ ($n\in {\mathbb Z}\setminus \{0\}$), the sums (4.1) are not dense in $L_p^0({\mathbb T})$. It is unclear whether this is true for $1\leqslant p<2$. For the uniform norm on the circle the following result holds. Theorem 4.3 (Borodin [11]). Let $f$ be a $2\pi$-periodic real-valued continuous function with Fourier expansion $\sum_{n\in {\mathbb Z}} c_ne^{int}$ such that (a) $c_0=0$ and $c_n\ne 0$ for $n\in {\mathbb Z}\setminus \{0\}$; (b) $\sum_{n\in \mathbb{Z}}|n|^2 |c_n|^2<\infty$ (that is, $f$ is absolutely continuous and $f'\in L_2(\mathbb{T})$). Then the sums (4.1) are dense in the real space with the uniform norm.It is unclear whether it is possible to replace condition (b) in Theorem 4.3 by the condition of the absolute continuity of $f$. In [11] it was shown that for any continuous function $f(t)=\sum_{n\in\mathbb{Z}\setminus \{0\}} c_n e^{int}$ satisfying $|c_n|\geqslant 1/|n|$, $n\in \mathbb{Z}\setminus \{0\}$ (such a function exists by a theorem of De Leeuw–Katznelson–Kahane: [48], Chap. 10, § 2) the sums (4.1) are not dense in $C_0(\mathbb{T})$. This, however, says nothing about the sharpness of condition (b) in Theorem 4.3. Theorems 4.1 and 4.3 cannot be transferred to the complex case. However, their natural analogues hold for spaces of holomorphic functions. Recall that the Hardy space $H_p({\mathbb T})$ ($1\leqslant p <\infty$) consists of those functions in the complex space $L_p({\mathbb T})$ whose Fourier coefficients with negative indices are equal to zero. This space is isometrically isomorphic to the Hardy space $H_p(|z|<1)$ of functions holomorphic in the unit disc. Theorem 4.4 (Borodin [11]). Let $1\leqslant p<\infty$, and let $f\in H_p({\mathbb T})$ be a function with Fourier expansion $\sum_{n=0}^\infty c_ne^{int}$ such that (a) $c_0=0$ and $c_n\ne 0$ for $n=1,2,\dots$; (b) $\sum_{n=1}^\infty n|c_n|^2<\infty$ for $1\leqslant p\leqslant 2$ or $\sum_{n=1}^\infty n|c_n|^q<\infty$ for $2 \leqslant p<\infty$ ($1/p+1/q=1$). Then the sums (4.1) are dense in the space The condition $\sum_{n=1}^\infty n|c_n|^2<\infty $ in (b) has the following geometric interpretation: the corresponding function $f(z)=\sum_{n=1}^\infty c_n z^n$ maps the unit disc onto a domain of finite area (taking into account the multiplicities of overlaps). The function
$$
\begin{equation*}
f(t)=\ln (1-e^{it})=\sum_{n=1}^\infty \biggl(-\frac{1}{n}\biggr)e^{int}
\end{equation*}
\notag
$$
is in $H_2^0(\mathbb{T})$ and satisfies condition (a) in Theorem 4.4, but the sums of its shifts are not dense in this space [11]. This example shows that condition (b) in Theorem 4.4 cannot be replaced by $|c_n|=O(1/n)$ ($n\to \infty$). An analogue of Theorem 4.3 holds for the space $\operatorname{AC}(\overline{U})$ of functions that are continuous on the closed disc $\overline{U}=\{z\colon |z|\leqslant 1\}$ and analytic in its interior. Theorem 4.5 (Borodin [11]). Let $f\in \operatorname{AC}(\overline{U})$ be a function with Taylor series $\sum_{n=0}^\infty c_nz^{n}$ such that (a) $c_0=0$ and $c_n\ne 0$ for $n=1,2,\dots$; (b) $\sum_{n=1}^\infty n^2 |c_n|^2<\infty$ ($f'$ belongs to the Hardy space $H_2$ in the disc). Then the sums are dense in the spaceThere are estimates for the rate of approximation by sums of shifts of one function on the circle [11]. Here is one of these estimates. Let $f_0$ be the $2\pi$-periodic function defined on $[0,2\pi)$ by It is easy to check that $f_0\in L_2^0({\mathbb T})$. If the function $h\in L_2^0({\mathbb T})$ has the integral modulus of continuity $\omega_2(h,\delta)\leqslant C\delta^\alpha$ for some $\alpha\in (0,1]$, then for every large $N$ there is a sum $S_q(t)=\displaystyle\sum_{j=1}^q\bigl(\pm f_0(t-a_j)\bigr)$ such that $q\leqslant N$ and
$$
\begin{equation*}
\|h-S_q\|\leqslant \frac{A}{N^{\alpha/(3-\alpha)}}\,,
\end{equation*}
\notag
$$
where $A$ is a constant depending only on $C$. It is unclear how sharp this estimate is. Theorem 4.5 can also be supplemented by a quantitative result, which is derived from Theorem 5 in [52]. Namely, if the Taylor coefficients of the function $f(z)=\sum c_n z^n\in \operatorname{AC}_0(\overline{U})$ satisfy for some $A>0$ and $s>0$, then any function $g(z)=\sum g_n z^n\in \operatorname{AC}_0(\overline{U})$ with coefficients $|g_n|=O(n^{-1-s})$ ($n\to\infty$) can be approximated by sums $\sum_{k=1}^N f(e^{ia_k}z)$, $a_k\in \mathbb{R}$, uniformly in the closed disc at a rate of $O(N^{-s})$.Of course, quantitative issues of approximations by sums of shifts of one function on the circle require further investigation. 4.2. Shifts on the lineTheorem 4.6 (Borodin [17]). There is a function $f\colon{\mathbb R}\to {\mathbb R}$ such that the sums of its shifts are dense in all real spaces $L_p({\mathbb R})$ for $2\leqslant p<\infty$ and also in the real space $C_0({\mathbb R})$. This result is essentially based on the theorem of Konyagin on the existence of trigonometric polynomials with positive integer coefficients converging to zero almost everywhere (Theorem 5.6 below). It is unclear whether Theorem 4.6 is valid for $1<p<2$. For $p=1$ it is obviously not valid, since shifts of one function do not form an all-round set in $L_1({\mathbb R})$: the functional $f\mapsto \displaystyle\int_{\mathbb R}f(x)\, dx$ takes the values of the same sign at all these shifts. In $L_\infty({\mathbb R})$, where an analogue of Theorem 4.6 is not valid either, the role of the forbidding functional is played by the Banach limit at $+\infty$ [17]. Theorem 4.6 fails in the complex spaces $L_p({\mathbb R})$ [17]; however, an analogue of it is valid for Hardy spaces in the half-plane. Theorem 4.7 (Dyuzhina [37]). There is a function $f$ defined in the upper half- plane such that the sums of real shifts of this function are dense in all Hardy spaces $H_p$ in the upper half-plane for $2\leqslant p<\infty$ and also in the space of functions that are analytic in the upper half-plane, continuous in its closure, and tend to zero at infinity. The proof of Theorem 4.7 repeats the proof of Theorem 4.6 on the level of ideas, but technically differs from it in many ways. The functions in Theorems 4.6 and 4.7 cannot be written out explicitly. However, if we approximate by a subgroup generated by plus-minus shifts of one function, then we can specify whole classes of functions for which such a subgroup is dense in the corresponding space. Theorem 4.8 (Borodin [9]). Suppose that $f$ is a function in the real space $L_2({\mathbb R})$, the Fourier transform $\widehat f$ vanishes only on a set of Lebesgue measure zero in ${\mathbb R}$, and the integral modulus of continuity satisfies $\omega^2_2(f,\delta)=o(\delta)$ as $\delta\to 0$ (for example, $f$ can be a Lipschitz function with compact support). Then the finite sums of functions $\pm f(t-\lambda)$, $\lambda\in {\mathbb R}$, are dense in $L_2({\mathbb R})$. The example $f=I_{[0,1]}$ shows that the conditions on the integral modulus of continuity of $f$ in Theorem 4.8 are essential and sharp (the subgroup of $L_2({\mathbb R})$ generated by this indicator function contains only integer-valued functions and is not dense in $L_2({\mathbb R})$, while $\omega_2^2(I_{[0,1]},\delta)=O(\delta)$). Interestingly, approximations by sums of plus-minus shifts of one function on the line are actively studied within the framework of so-called $\Sigma\Delta$-quantization. Here is one typical result [34]. Suppose $\lambda>1$ and the function $g$ is the Fourier transform of an infinitely differentiable function which has its support in $[-\lambda \pi,\lambda\pi]$ and is equal to $1/\sqrt{2\pi}$ on $[-\pi, \pi]$. Then for every $k\in {\mathbb N}$ and every function $f$ that is the Fourier transform of a finite measure with support in $[-\pi,\pi]$ and satisfies $\|f\|_\infty<1$, there are effectively calculated $q_n^{(k)}\in\{-1,1\}$ such that
$$
\begin{equation*}
\biggl\|f(t)-\frac{1}{\lambda}\sum_n q_n^{(k)} g\biggl(t-\frac{n}{\lambda}\biggr)\biggr\|_\infty\leqslant \frac{C(g,k)}{\lambda^k}\,.
\end{equation*}
\notag
$$
As one can see, the sizes of possible shifts are additionally quantized here. Komarov [54] obtained the following result on approximation by sums of complex plus-minus shifts of the function $1/x$ (that is, by differences of simple partial fractions), related to the above in a certain sense. Let the real-valued function $f\in C_0({\mathbb R})\cap L_1({\mathbb R})$ satisfy the Lipschitz condition of order $\alpha\in (0,1)$, and let its Hilbert transform satisfy $Hf\in L_1({\mathbb R})$. Then for every $n=2,3,\dots$ there are efficiently defined sets $\{a_1,\dots,a_{2n}\}$ and $\{b_1,\dots,b_{2n}\}$ of complex numbers, which are symmetric with respect to the real line, and such that
$$
\begin{equation*}
\biggl\|f(t)-\frac{\|Hf\|_{L_1({\mathbb R})}}{4\pi n} \biggl(\,\sum_{j=1}^{2n} \frac{1}{t-a_j}- \sum_{j=1}^{2n} \frac{1}{t-b_j}\biggr)\biggr\|_\infty\leqslant \frac{C(f)}{n^{\alpha/(1+\alpha)}}\,.
\end{equation*}
\notag
$$
In connection with these results, it would be interesting to obtain estimates for the rate of approximation of functions in some classes by sums of arbitrary plus-minus shifts of one function from Theorem 4.8. 4.3. Shifts on a latticeIn spaces of two-sided sequences $x=(\dots,x_{-1},x_0, x_1,x_2,\dots)$, the (right) shift operator $T$ is defined: $(Tx)_n=x_{n-1}$, $n\in {\mathbb Z}$. Theorem 4.9 (Borodin [12]). There is an element $v$ in the real space $\ell_2({\mathbb Z})$ of two-sided sequences such that the finite sums $\displaystyle\sum T^{n_k}v$ of its shifts are dense in all real spaces $\ell_p({\mathbb Z})$, $2\leqslant p< \infty$, and also in the real space $c_0({\mathbb Z})$. This result is also essentially based on a theorem of Konyagin (Theorem 5.6 below). It is unclear whether it is valid for $1<p<2$. In the spaces $\ell_1({\mathbb Z})$, $\ell_\infty({\mathbb Z})$ and $c({\mathbb Z})$ (two-sided sequences having a limit in both directions, with the uniform norm), the shifts of one vector form a set which is not all-round, and their sums are not dense [12]. The proof of Theorem 4.9 does not allow us to present a concrete vector whose sums of shifts are dense in real $l_2({\mathbb Z})$. The problem of constructing such a vector remains open. It is not so easy even to come up with a non-zero vector for which sums of shifts have arbitrarily small norms. The following assertion provides an example of such a vector and describes the subspace of $l_2({\mathbb Z})$ ’filled’ by the sums of shifts of this vector. Theorem 4.10 (Borodin [12]). The sums of shifts of the vector (which represents the sequence of trigonometric Fourier coefficients of the function $\sqrt{\pi/2}\,I_{[-2\pi/3,-\pi/3]\cup[\pi/3,2\pi/3]}$) converge to zero in $l_2({\mathbb Z})$. The closure of the set of all sums of shifts of this vector is the linear subspace of Fourier coefficients of those functions in $L_2({\mathbb T})$ that have real Fourier coefficients and have support on $[-2\pi/3,-\pi/3]\cup[\pi/3,2\pi/3]$. 4.4. Possible generalizationsDyuzhina [39] generalized Theorems 4.1, 4.6, and 4.9 to the multidimensional case. Namely, the sums of shifts of any fixed function which has rather rapidly decreasing non-zero Fourier coefficients are dense in all real spaces $L_p^0(\mathbb{T}^d)$ of functions with zero mean value on the $d$-dimensional torus, $p\in (1,\infty)$. In the real spaces $L_p(\mathbb{R}^d)$ and $l_p(\mathbb{Z}^d)$ for $2\leqslant p<\infty$, there is, respectively, a function and a vector with dense sums of shifts. Spaces $L_p$ and shifts of function are defined on every locally compact Abelian group. Therefore, the results formulated above lead naturally to the problem of the description of locally compact Abelian groups $G$ on which there are functions whose sums of shifts are dense in $L_2(G)$ (or in $L_2^0(G)$ if $G$ is compact). Here $L_2(G)$ denotes the space of real functions that are square summable on $G$ with respect to Haar measure. An attempt to generalize Theorem 4.8 on the density of sums of plus-minus shifts of one function to the case of locally compact groups was undertaken in [75], but the statement formulated in that paper (Corollary 2.11) is incorrect. Of course, one need not limit oneself to locally compact groups, but can consider the problem on the density of the sums of shifts of one function on any set where there are plenty of these shifts, for example, on a $d$-dimensional sphere. In ${\mathbb R}^d$, not only shifts can be considered, but also dilations and, in general, arbitrary families of linear transformations of variables. For example, given a function $f$ of one variable, one can state the problem on the density of sums
$$
\begin{equation*}
\sum_{k=1}^n f(w_k\cdot x-a_k), \qquad x\in E\subset {\mathbb R}^d,\quad w_k\in W\subset {\mathbb R}^d,\quad a_k\in A\subset {\mathbb R}^d
\end{equation*}
\notag
$$
(sums, with coefficients equal to 1, of ridge functions generated by one function), in various spaces of functions defined on the set $E$. Here is one recent result on approximation by sums of shifts and dilations of one function. Theorem 4.11 (Filippov [43]). Let $1\leqslant p< \infty$, and let $\psi\colon{\mathbb R}\to {\mathbb R}$ be a function with support on $[0,1]$ such that $\psi\in L_p[0,1]$ and $\displaystyle\int_0^1 \psi(t)\,dt\ne0$. Set Then every function $g\in L_p[0,1]$ admits an expansion $\sum_n a_n\psi_n$ with integer coefficients, which converges to $g$ in $L_p[0,1]$. We have formulated here a one-dimensional result for simplicity; in [43] the corresponding assertion is proved for the $d$-dimensional cube $[0,1]^d$, where $d$ is arbitrary. Curiously, in the Hilbert case $p=2$ Theorem 2.27 gives another expansion of an arbitrary function $g\in L_2[0,1]$, namely, the expansion $\displaystyle\sum \varepsilon_k \psi_{n_k}$, where $\varepsilon_k\in\{\pm 1\}$, and the indices $n_k$ can repeat. Indeed, let us show that the set $\{\pm \psi_n\}$ reduces the norm in $L_2[0,1]$. Otherwise there is a non-zero function $g\in L_2[0,1]$ such that
$$
\begin{equation}
\|g\pm \psi_n\|^2\geqslant \|g\|^2\quad\Longleftrightarrow\quad |\langle g, \psi_n \rangle|\leqslant \frac{\|\psi_n\|^2}{2}= \frac{\|\psi\|^2}{2^{3k+1}}
\end{equation}
\tag{4.4}
$$
for all $n$. We set $\delta=\displaystyle\int_0^1 \psi(t)\,dt\ne0$. For every interval
$$
\begin{equation*}
\Delta_n=\operatorname{supp}\psi_n= \biggl[\frac{j}{2^k}\,,\frac{j+1}{2^k}\biggr]
\end{equation*}
\notag
$$
we have
$$
\begin{equation}
\begin{aligned} \, \nonumber \int_{\Delta_n}g(t)\delta\,dt&=\int_{\Delta_n}g(t)(\delta-2^k\psi_n(t))\,dt+ \int_{\Delta_n} g(t)\,2^k\psi_n(t)\,dt \\ &=\int_{\Delta_n} (g(t)-g_n)(\delta-2^k\psi_n(t))\,dt+ \int_{\Delta_n} g(t)\,2^k\psi_n(t)\,dt, \end{aligned}
\end{equation}
\tag{4.5}
$$
where is the mean value of $g$ on $\Delta_n$. Let $\Delta\subset[0,1]$ be an arbitrary closed binary-rational interval. For sufficiently large $k$, $\Delta$ is tiled by closed intervals $\Delta_n$, $n=2^k+j$ for fixed $k$. Denoting the $1/2^k$-periodic extension of the function $\psi(2^kt)$ on $[0,1]$ by $\Psi_k$ and summing equalities (4.5) for all selected $n$ we obtain
$$
\begin{equation*}
\delta\int_{\Delta} g(t)\,dt=\int_{\Delta}\biggl(g(t)- \sum_n g_nI_{\Delta_n}(t)\biggr)(\delta-\Psi_k(t))\,dt+ \sum_n\int_{\Delta_n} g(t)\,2^k\psi_n(t)\,dt.
\end{equation*}
\notag
$$
The last sum does not exceed $\|\psi\|^2/2^{k+1}$ in view of (4.4), while the first summand does not exceed
$$
\begin{equation*}
\biggl\|g-\sum_n g_nI_{\Delta_n}\biggr\|_{L_2(\Delta)}\cdot \|\delta- \Psi_k\|_{L_2(\Delta)}\to 0 \qquad (k\to \infty),
\end{equation*}
\notag
$$
since
$$
\begin{equation*}
\|\delta-\Psi_k\|^2_{L_2(\Delta)}\leqslant \|\delta-\Psi_k\|^2_{L_2[0,1]}=\|\psi\|^2-\delta^2.
\end{equation*}
\notag
$$
Thus, $\displaystyle\int_{\Delta} g\,dt=0$ for every binary-rational $\Delta$, hence $g=0$, which is a contradiction. Theorems 4.11 and 2.27 provide two different methods for expansion in a series in $L_2[0,1]$ with respect to the system $\{\psi_n\}$. It would be interesting to compare these methods in terms of the rates of convergence using the model example $\psi=I_{[0,1]}$. On the other hand there is an additional motivation to generalize Theorem 2.27 to Banach spaces. We present another new result generalizing Theorem 4.1 to the case of a two- dimensional torus. It is interesting primarily because it provides a positive solution to Problem 2.16 in a particular case. Theorem 4.12 (Borodin). Suppose a real-valued function $f(x,y)$, which is defined on the two-dimensional torus $\mathbb{T}^2=[0,2\pi)\times [0,2\pi)$ and extended to the whole plane $2\pi$-periodically in both variables, has the Fourier expansion such that (a) $c_{00}=0$ and $c_{mn}\ne 0$ for $m^2+n^2>0$; (b) $\sum_{m,n\in \mathbb{Z}}(m^2+n^2)|c_{mn}|^2<\infty$ (that is, $f$ has the Lipschitz modulus of continuity in $L_2(\mathbb{T}^2)$). Then the sums are dense in the real spaceCondition (b) here is weaker than the corresponding condition
$$
\begin{equation*}
\sum_{m,n\in \mathbb{Z}}(\max\{m,n\})^3 |c_{mn}|^2<\infty
\end{equation*}
\notag
$$
in [39]. However, in [39] sums of shifts are considered in $L_p^0(\mathbb{T}^d)$ for arbitrary $p\in (1,\infty)$ and $d\in {\mathbb N}$. Proof. 1. It is sufficient to approximate the function zero by sums (4.6), that is, for every $\varepsilon>0$ to choose $\zeta_k=e^{i\alpha_k}$ and $\xi_k=e^{i\beta_k}$ so that
$$
\begin{equation}
\biggl\|\,\sum_{k=1}^N f(x+\alpha_k,y+\beta_k)\biggr\|= \sum_{m,n\in \mathbb{Z}}\biggl|\,\sum_{k=1}^N \zeta_k^m\xi_k^n\biggr|^2 |c_{mn}|^2<\varepsilon.
\end{equation}
\tag{4.7}
$$
Indeed, the pair $\alpha_1$, $\beta_1$ in (4.7) can be considered arbitrary prescribed (all $\alpha_k$, $\beta_k$ can be shifted similarly without violating inequality (4.7)), that is, any minus-shift $-f(x+\alpha,y+\beta)$ can be approximated by sums (4.6) with arbitrary accuracy. This means that the closure of the semigroup consisting of sums (4.6) is a subgroup of $L_2^0(\mathbb{T}^2)$. This semigroup is generated by the set which is a Lipschitz image of a square (condition (b) is equivalent to the fact that the $L_2$-modulus of continuity of $f$ satisfies $\omega_2(f,\delta)=O(\delta)$ as $\delta\to 0$). We show that this set is all-round in $L_2^0(\mathbb{T}^2)$. For a function $g\in L_2^0(\mathbb{T}^2)$ with Fourier coefficients $g_{mn}$ the inequality is equivalent to The series in the variables $\alpha$ and $\beta$ on the left-hand side of this inequality converges absolutely to a continuous function with zero mean value. Therefore, this function is identically zero, hence $c_{mn} \overline{g_{mn}}=0$, which in view of (a) implies $g_{mn}=0$ for all $m,n$.
Thus, once (4.7) has been proved, Theorem 2.18 provides the density of the sums (4.6) in $L_2^0(\mathbb{T}^2)$. 2. For $\zeta_k=\omega^{u(k-1)}$ and $\xi_k=\omega^{v(k-1)}$, where $\omega=e^{2\pi i/N}$, while $u$ and $v$ are some positive integers, we have
$$
\begin{equation*}
\sum_{k=1}^N\zeta_k^m\xi_k^n=\sum_{k=0}^{N-1}(\omega^{um+vn})^k\begin{cases} N, & um+vn \ \text{is a multiple of} \ N, \\ 0, & um+vn \ \text{is not a multiple of}\ N, \end{cases}
\end{equation*}
\notag
$$
so that the expression in (4.7) is equal to
$$
\begin{equation}
N^2\sum |c_{mn}|^2=N^2\sum\nolimits_{\rm I}|c_{mn}|^2+ N^2\sum\nolimits_{\rm II} |c_{mn}|^2,
\end{equation}
\tag{4.8}
$$
where the left-hand sum is taken over all $m,n\in {\mathbb Z}$ such that $um+vn$ is a multiple of $N$, the sum $\displaystyle\sum\nolimits_{\rm I}$ is taken over all $m,n\in {\mathbb Z}$ such that $um+vn=0$, and the sum $\displaystyle\sum\nolimits_{\rm II}$ is taken over all $m,n\in {\mathbb Z}$ such that $um+vn\ne0$ and $um+vn$ is a multiple of $N$.
3. To estimate the sum $\displaystyle\sum\nolimits_{\rm I}$ in (4.8) we consider the following series:
$$
\begin{equation*}
\begin{aligned} \, \sum_{(u,v)=1}(u^2+v^2)\sum_{um+vn=0}|c_{mn}|^2&= \sum_{m,n\in{\mathbb Z}\colon mn<0} \biggl(\biggl(\frac{m}{(m,n)}\biggr)^2+ \biggl(\frac{n}{(m,n)}\biggr)^2\biggr)|c_{mn}|^2 \\ &<\sum_{m,n\in \mathbb{Z}}(m^2+n^2) |c_{mn}|^2<\infty. \end{aligned}
\end{equation*}
\notag
$$
On the other hand
$$
\begin{equation*}
\begin{aligned} \, \sum_{(u,v)=1}\frac{1}{u^2+v^2}&\geqslant \sum_{p\in \mathbb{P}}\, \sum_{u \colon p\nmid u}\frac{1}{u^2+p^2} \\ &=\sum_{p\in \mathbb{P}}\biggl(\,\sum_{u=1}^\infty\frac{1}{u^2+p^2}- \sum_{q=1}^\infty\frac{1}{p^2(q^2+1)}\biggr)\geqslant \sum_{p\in \mathbb{P}}\biggl(\frac{1}{p}-\frac{2}{p^2}\biggr)=\infty, \end{aligned}
\end{equation*}
\notag
$$
where $\mathbb{P}$ denotes the set of prime numbers. Consequently, for every $\varepsilon>0$ and every $R>0$ there is a pair of mutually prime itegers $u$ and $v$ such that $u^2+v^2>R$ and
$$
\begin{equation*}
(u^2+v^2)^2\sum_{um+vn=0}|c_{mn}|^2< \frac{\varepsilon}{2}\,.
\end{equation*}
\notag
$$
For these mutually prime $u$ and $v$ and each $N\leqslant u^2+v^2$ the first sum $\displaystyle\sum\nolimits_{\rm I}$ on the right-hand side of (4.8)) is less than $\varepsilon/2$.
4. We find $N\leqslant u^2+v^2$ such that the second sum $\displaystyle\sum\nolimits_{\rm II}$ on the right-hand side of (4.8) is small too. We have
$$
\begin{equation}
\begin{aligned} \, \nonumber \sum_{N=(u^2+v^2)/2}^{u^2+v^2}N^2\displaystyle\sum\nolimits_{\rm II}|c_{mn}|^2&\leqslant \sum_{m,n\colon |um+vn|\geqslant (u^2+v^2)/2} \biggl(\,\sum_{N|um+vn} N^2\biggr)|c_{mn}|^2 \\ &\leqslant \sum_{m,n\colon|um+vn|\geqslant (u^2+v^2)/2}2(um+vn)^2|c_{mn}|^2 \end{aligned}
\end{equation}
\tag{4.9}
$$
(we have used the fact that the sum of squares of divisors of $K$ does not exceed $K^2+(K/2)^2+(K/3)^2+\dots\leqslant 2K^2$). The right-hand side of (4.9) is bounded by
$$
\begin{equation*}
\begin{aligned} \, &2(u^2+v^2)\sum_{m,n\colon |um+vn|\geqslant (u^2+v^2)/2}(m^2+n^2)|c_{mn}|^2 \\ &\qquad\leqslant 2(u^2+v^2)\sum_{m,n\colon m^2+n^2\geqslant (u^2+v^2)/4} (m^2+n^2) |c_{mn}|^2 \\ &\qquad\leqslant 2(u^2+v^2)\sum_{m,n\colon m^2+n^2>R/4}(m^2+n^2) |c_{mn}|^2\leqslant \frac{u^2+v^2}{2}\, \frac{\varepsilon}{2} \end{aligned}
\end{equation*}
\notag
$$
for sufficiently large $R$. Hence there is $N$ between $(u^2+v^2)/2$ and $u^2+v^2$ such that
$$
\begin{equation*}
N^2\displaystyle\sum\nolimits_{\rm II} |c_{mn}|^2<\frac{\varepsilon}{2}\,.
\end{equation*}
\notag
$$
For the selected $u$, $v$, and $N$, the left-hand side in (4.8) is less than $\varepsilon$. $\Box$ It would be interesting to generalize Theorem 4.12 to the multidimensional case. It is also unclear how sharp the condition (b) in this theorem is: differences of indicators of disjoint rectangles of the same area in ${\mathbb T}^2$ forbid only the condition $\sum_{m,n\in \mathbb{Z}}(m^2+n^2)^\gamma|c_{mn}|^2<\infty$, with $\gamma<1/2$. 5. Density of polynomials with integer coefficientsApproximation by polynomials with integer coefficients in various function spaces is an extensive topic rich in deep and subtle results that can be found in [46], [77], [42], and [62], Chap. 2. These results essentially use the algebraic properties of polynomials. Meanwhile, it is quite natural to set the problem of approximation by linear combinations with integer coefficients with respect to an arbitrary system. First of all, the question arises about the density of such combinations. Problem 5.1. (i) Let $\{u_n\}_{n=1}^\infty$ be a complete system in a Banach space $X$. Find conditions that are necessary or sufficient for the subgroup to be dense in $X$. (ii) Let $\{u_n\}_{n=1}^\infty$ be an all-round system in a Banach space $X$. Find conditions that are necessary or sufficient for the semigroup to be dense in $X$.For example, it would be interesting to obtain results in the framework of the above general problem that would imply the following well-known theorem. Theorem 5.2 (Aparicio Bernardo [3], see also [46]). Algebraic polynomials with integer coefficients are dense in the real space $L_2(\Delta)$ on an interval $\Delta\subset {\mathbb R}$ if and only if the length of $\Delta$ is less than 4. This theorem is essentially used in the proof of Theorem 4.9. Here are some primary observations in connection with Problem 5.1. Theorem 5.3 (Borodin and Shklyaev). Let $\{u_n\}_{n=0}^\infty$ be a complete system in a Banach space $X$, let $Y_n=\operatorname{span}\{u_0,\dots,u_{n-1}\}$ for $n\geqslant 1$ and $Y_0=\{0\}$. (1) If the subgroup $R(\{\pm u_n\})$ is dense in $X$, then (2) If $\sum_{n=0}^\infty\operatorname{dist}(u_n,Y_n)<d$, then for every $x\in X$ the inequality holds.(3) If $X$ is uniformly smooth with modulus of smoothness $s(\,\cdot\,)$, $\sum_{n=0}^\infty s(\|u_n\|)<\infty$, and the system $\{u_n\}_{n=m}^\infty$ is complete in $X$ for every $m$, then the subgroup $R(\{\pm u_n\})$ is dense in $X$. (4) If $X$ is uniformly smooth with modulus of smoothness $s(\,\cdot\,)$, $\sum_{n=0}^\infty s(\|u_n\|)<\infty$, and the system $\{u_n\}_{n=m}^\infty$ is all-round in $X$ for every $m$, then the semigroup $R(\{u_n\})$ is dense in $X$. Proof. (1) Let $\varepsilon>0$ and $k\in {\mathbb N}$. We take an element $x$ such that $\|x\|=\varepsilon=\operatorname{dist}(x,Y_k)$ and approximate it to within $\varepsilon/2$ by an element $u\in R(\{\pm u_n\}))$. Clearly, $u\notin Y_k$, that is, in the representation of $u$ as a finite combination of the elements $\{u_j\}$ with integer coefficients, for the element $u_n$ with largest index we have $n\geqslant k$. For this $n$ we have $\operatorname{dist}(u_n,Y_n)\leqslant \|u\|\leqslant 3\varepsilon/2$.
(2) The idea of the following arguments goes back to Kakeya [67]. Set Consider an arbitrary element $x\in X$ and a linear combination $\lambda_nu_n+\dots+\lambda_0u_0$ approximating this element. Subtracting $(\lambda_n-[\lambda_n])p_n$, we obtain the new combination $\lambda_n^1u_n+\dots+\lambda_0^1u_0$ with integer $\lambda_n^1=[\lambda_n]$. Subtracting $(\lambda_{n-1}^1-[\lambda_{n-1}^1])p_{n-1}$ from it, we obtain a combination with two integer coefficients. Continuing in this way we obtain an integer combination different from the original one by at most
$$
\begin{equation*}
\sum_{k=0}^n\|p_k\|=\sum_{k=0}^n\operatorname{dist}(u_k,Y_k)<d.
\end{equation*}
\notag
$$
Since the original combination approximates $x$ with arbitrary accuracy, we conclude that $\operatorname{dist}(x,R(\{\pm u_n\}))<d$.
(3) Let $m$ be a positive integer such that $\sum_{n=m}^\infty s(\|u_n\|)<1$. Consider an arbitrary element $x\in X$ and a linear combination
$$
\begin{equation*}
u=\lambda_mu_m+\dots+\lambda_nu_n,\qquad n>m,\quad \|x-u\|<\varepsilon,
\end{equation*}
\notag
$$
which approximates this element. We set (here $\{\,\cdot\,\}$ denotes the fractional part of the number). By Lemma 2.3 there is a combination $u''=\theta_mu_m+\dots+\theta_nu_n$ such that $\theta_j\in \{0,1\}$ and
$$
\begin{equation*}
\|u'-u''\|\leqslant A\biggl(\,\max_{m\leqslant j\leqslant n}\|u_j\|+ \biggl(\,\sum_{j=m}^n s(\|u_j\|)\biggr)^\gamma\biggr),
\end{equation*}
\notag
$$
where the constants $A$ and $0<\gamma\leqslant 1$ depend only on the function $s(\tau)$. Since $s(\|u_k\|)\to 0$, we also have $\|u_k\|\to 0$, so that both summands on the right-hand side tend to zero as $m\to\infty$, and $\|u'-u''\|<\varepsilon$ for large $m$. Consequently, $\|x-(u-u'+u'')\|< 2\varepsilon$ and, in addition, $u-u'+u''\in R(\{\pm u_n\})$, hence $ R(\{\pm u_n\})$ is dense $X$.
(4) This statement is proved similarly to (3), except that, at the first step, one has to use Lemma 2.1 to select an approximating linear combination $u=\lambda_mu_m+\dots+\lambda_nu_n$ with non-negative coefficients. $\Box$ Remark 5.4. The condition $\sum_{n=0}^\infty s(\|u_n\|)<\infty$ in statement (3) of Theorem 5.3 cannot be replaced by $\sum_{0}^\infty\operatorname{dist}(u_n,Y_n)<\infty$. Indeed, consider the following example in a Hilbert space with orthonormal basis $\{e_n\}_{n=0}^\infty$. We set where the sequence $p_n$ runs infinitely many times through each element $e_k/2^k$, for example, in the following order: Clearly, $Y_n=\operatorname{span}\{e_0,\dots,e_{n-1}\}$, $\operatorname{dist}(u_n,Y_n)=1/2^n$, while the closure of any subsystem $\{u_n\}_{n=m}^\infty$ as a set contains all elements $e_k/2^k$, so that this subsystem is complete. At the same time all elements of the subgroup $R(\{\pm u_n\})$ have quantized Fourier coefficients: the $k$th coefficient is always a multiple of $1/2^k$.The following remark shows a certain sharpness of statement (3) in Theorem 5.3. Remark 5.5. For every sequence there is a system $\{u_n\}_{n=0}^{\infty}$ in a Hilbert space $H$ such that $\|u_n\|^2=\alpha_n$ for all $n$, the system $\{u_n\}_{n=m}^\infty$ is complete in $H$ for every $m$, but the subgroup $R(\{\pm u_n\})$ is not dense in $H$. Indeed, let us take as $H$ the space $L_2$ on the circle ${\mathbb T}$. We arrange a sequence of arcs $\Delta_n$ with lengths $|\Delta_n|=\alpha_n$ on ${\mathbb T}$ by attaching each arc to the previous one in one direction. We set $u_n=I_{\Delta_n}$. Clearly, $\|u_n\|^2=\alpha_n$. That any subsystem $\{u_n\}_{n=m}^\infty$ is complete follows from the fact that the closure of its linear span contains the indicator function $I_\Delta$ of each interval $\Delta\subset {\mathbb T}$: when going around the circle, a properly selected union $\bigcup_{n=k}^l\Delta_n$ of consecutive arcs differs arbitrarily little from $\Delta$ because $\alpha_n\to 0$. At the same time, it is clear that the subgroup $R(\{\pm u_n\})$ consists only of integer-valued functions and is not dense in $L_2({\mathbb T})$. Statement (1) of Theorem 5.3 entails necessity in Theorem 5.2: the Legendre polynomials for an interval $[a,b]$, which form an orthonormal basis in $L_2[a,b]$, have the form
$$
\begin{equation*}
L_n(x)=\frac{\sqrt{2n+1}\,(2n)!}{(b-a)^{n+1/2}(n!)^2}\,x^n+\cdots,
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
\operatorname{dist}(x^n,\operatorname{span}\{1,x,\dots,x^{n-1}\})= \frac{(b-a)^{n+1/2}(n!)^2}{\sqrt{2n+1}\,(2n)!} \asymp \biggl(\frac{b-a}{4}\biggr)^n, \qquad n\to\infty.
\end{equation*}
\notag
$$
On the other hand, statement (3) of Theorem 5.3 entails the density of polynomials with integer coefficients in $L_2[0,1]$ (one must take $u_0=1$ and $u_n=t^n-t^{n-1}$, $n=1,2,\dots$). We present another result on trigonometric polynomials with positive integer coefficients, which is essentially used in the proofs of Theorems 4.6, 4.7, and 4.9. Theorem 5.6 (Konyagin [17]). There exists a sequence of trigonometric polynomials converging to zero almost everywhere, where the $k_s^{(\nu)}$ are integers and the $n_s^{(\nu)}$ are positive integers. This very difficult theorem raises the natural question on the existence of a sequence of trigonometric polynomials with positive integer coefficients that would converge to zero in the $L_2$-norm or even uniformly on each interval $[\delta,2\pi-\delta]$, $\delta\in(0,\pi)$. A positive answer to this question would allow us to extend Theorems 4.6 and 4.9 to all values $p\in (1,\infty)$. Note that there is a sequence of trigonometric polynomials with integer coefficients that converges uniformly to zero on these intervals, which follows from some results due to Fekete [41]. The following more specific question can be posed. Problem 5.7. Is it true that for every $\delta>0$ there is a polynomial where the $k_s$ are integers and the $n_s$ are positive integers, such that $|Q_\delta(x)| < 1$ for $x\in [\delta, 2\pi-\delta]$? For $\delta=1.35$ such a polynomial, namely, was found by Dyuzhina. In the general case such a polynomial $Q_\delta$ would provide a sequence $Q_\delta^n$ of trigonometric polynomials converging uniformly to zero on $[\delta,2\pi- \delta]$.The authors are grateful to A. R. Alimov, N. A. Dyuzhina, B. S. Kashin, M. A. Komarov, and Yu. A. Skvortsov for valuable comments. 
Citation in format AMSBIB
\Bibitem{BorShk23} |
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