| Sbornik: Mathematics |
|
|
|
|
|
On the density of the additive semigroup generated by a subset of a Hilbert–Schmidt ellipsoid K. S. Shklyaevab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
Bibliographic databases:
    § 1. IntroductionLet $X$ be a real Banach space and $M$ be a subset of $X$. The set
$$
\begin{equation*}
\mathcal{R}(M)=\bigcup_{n=1}^\infty \underbrace{M+\dots+M}_{n}
\end{equation*}
\notag
$$
consisting of all possible sums of elements of $M$ is called the additive semigroup generated by $M$. The study of additive semigroups was pioneered by Borodin. In particular, he stated the following problem. Problem 1. Characterize the sets $M$ for which the additive semigroup $\mathcal{R}(M)$ is dense in the Banach space $X$. In its general form Problem 1 was first considered in [1], even though various particular cases of it, namely, approximation by polynomials with integer or natural integer coefficients or approximation by simple partial fractions $\sum_{i=1}^n 1/(z - w_i)$, were investigated long before [1]. For advances in Problem 1, see the recent survey [2]. Note that if the additive semigroup is dense, then its generating set is all-round. Recall that $M$ is an all-round set if it is not contained in any closed half-space $\{{x \in X}\colon f(x) \geqslant 0\}$, $f \in X^\ast \setminus \{0\}$. Indeed, if $M$ is not all-round, that is, $M$ lies in some half-space $\{x \in X\colon f(x) \geqslant 0\}$, then the additive semigroup $\mathcal{R}(M)$ is also contained in this half-space, and so this semigroup is not dense in $X$. However, mere all-roundedness is insufficient for an additive semigroup to be dense even in the finite-dimensional setting. So, for example, the integer lattice $\mathbb{Z}^n$ is an all-round set in $\mathbb{R}^n$ but, clearly, $\mathcal{R}(\mathbb{Z}^n)=\mathbb{Z}^n$. For the additive semigroup $\mathcal{R}(M)$ generated by the set $M$ to be dense in $X$ this semigroup should satisfy the following stronger condition of weak density. Definition 1. Let $X$ be a Banach space. An additive semigroup $\mathcal{R}(M)$ is weakly dense in $X$ if for each functional $f \in X^\ast \setminus \{0\}$ the set $f(\mathcal{R}(M)):= \{f(x)\colon {x \,{\in}\, \mathcal{R}(M)}\}$ is dense in $\mathbb{R}$. In a finite-dimensional space weak density is equivalent to (norm) density (see Proposition 1 below). However, in the infinite-dimensional setting the weak density of an additive semigroup does not generally imply its density — for example, the set of indicator functions $M=\{\pm I_{[0,t]}\colon t \in [0,1]\}$ is all-round and connected in $L_2[0,1]$, and so it is weakly dense in this space (see Proposition 3 below). It is clear that $\overline{\mathcal{R}(M)}$ contains only functions that assume integer values almost everywhere, but these functions are not dense in $L_2[0,1]$. In the earlier results on the density of semigroups it was typically assumed that the set generating the semigroup, in addition to being all-round, must satisfy quite stringent geometric conditions of rectifiability type. We recall one result from [2]. Theorem A. Let $X$ be a uniformly smooth Banach space and $\Gamma_i\colon [0,1] \to X$, $i=1,\dots,m$, be rectifiable curves such that the set $\Gamma:= \bigcup_{i=1}^m \Gamma_i([0,1])$ is all-round in $X$. Then the following hold. (1) If $m=1$, then $\overline{\mathcal{R}(\Gamma)}=X$. (2) If $m>1$, then $\overline{\mathcal{R}(\Gamma)}$ is a closed additive subgroup of $X$ containing a subspace $L$ of real codimension $m-1$ and such that each functional $f\in L^\perp$ (an annihilator in the realified space $X^\ast$) is constant on each curve $\Gamma_i$, $i=1,\dots,m$. In the present paper we obtain a theorem in which the density of an additive semigroup generated by a set $M$ is claimed under the assumption that this set lies in a Hilbert–Schmidt ellipsoid (also called an $s$-ellipsoid) — this is an ellipsoid for which the sum of squared semiaxes is finite. This condition replaces a rectifiability type condition for the set $M$. We show that the image of a one-dimensional compact set under a Lipschitz mapping is contained in an $s$-ellipsoid; however, if $d\geqslant 3$, then this claim may fail to hold for a $d$-dimensional compact set. In particular (see Example 2.17 in [2]), there exist an all-round set $M$ in a Hilbert space $H$ that is the image of a curve in $\mathbb{R}^3$ under a Lipschitz mapping, but the semigroup $\mathcal{R}(M)$ generated by this set is not dense in $H$. Nevertheless, it turns out that the image of any finite-dimensional compact set under a holomorphic map to a complex Hilbert space is contained in an $s$-ellipsoid. So the density criterion for an additive semigroup generated by a subset of an $s$-ellipsoid applies to a broad class of additive semigroups. The paper is organized as follows. In § 2 we present various necessary and sufficient conditions for an additive semigroup to be dense in the space; in § 3 we prove Theorem 1 on approximation by an additive semigroup generated by a subset of an $s$-ellipsoid. In § 4 we present sufficient conditions for a set to lie in an $s$-ellipsoid. As a consequence, we derive Theorem 2 on approximation by translations of a holomorphic function; this result is an extension of Korevaar’s theorem on approximation by simple partial fractions. § 2. Weak density of a semigroupProposition 1. An additive semigroup $\mathcal{R}(M)$ in a finite-dimensional normed space $X$ is weakly dense if and only if it is dense in $X$. Proof. If $\mathcal{R}(M)$ is weakly dense, then $M$ is an all-round set, and so $\overline{\mathcal{R}(M)}$ is an additive subgroup by Theorem 1 in [1]. The closed additive subgroup $\overline{\mathcal{R}(M)}$ of a finite-dimensional space is the sum of an $\mathbb{R}$-linear subspace $Y$ and a discrete lattice $L$, where $Y \cap L=\{0\}$ (see Ch. 7, § 1 in [3]). Let $e_1, \dots, e_k$ be a basis of the subspace $Y$ and $e_{k+1}, \dots, e_n$ be a basis of the lattice $L$. Assume that $\mathcal{R}(M)$ is not dense in $X$. Then $k < n$. It is clear that $n \leqslant \dim X$. Hence there exists a nonzero functional $f \in X^\ast$ such that $f(e_1)=0$, $\dots$, $f(e_k)=0$ and $f(e_{k+1})=1$, $\dots$, $f(e_{n})=1$. But in this case $f(\mathcal{R}(M))=\mathbb{Z}$, and so the semigroup $\mathcal{R}(M)$ is not weakly dense in $X$.
This proves Proposition 1. Proposition 2. An additive semigroup $\mathcal{R}(M)$ in a Banach space $X$ is weakly dense if and only if it is dense in the weak topology of $X$. Proof. It suffices to verify that if $\mathcal{R}(M)$ is weakly dense in $X$, then $\mathcal{R}(M)$ is dense in the weak topology of $X$. We claim that if $\varepsilon > 0$, $f_1, \dots, f_n \in X^\ast$ are linearly independent functionals and $c_1, \dots, c_n \in \mathbb{R}$, then there exists an element $y \in \mathcal{R}(M)$ such that $|f_i(y) - c_i| < \varepsilon$, $i=1, \dots, n$. Consider the map $F\colon \mathcal{R}(M) \to \mathbb{R}^n$ defined by $x \mapsto (f_1(x), \dots, f_n(x))$. It is clear that $F(\mathcal{R}(M))$ is an additive semigroup in $\mathbb{R}^n$. If $\overline{F(\mathcal{R}(M))} \neq \mathbb{R}^n$, then by Proposition 1 there exists a nontrivial functional $\lambda=(\lambda_1, \dots, \lambda_n) \in \mathbb{R}^n$ such that the set $\langle F(\mathcal{R}(M)), \lambda\rangle=\bigl\{\sum_{i=1}^n \lambda_i f_i(x)\colon x \in \mathcal{R}(M)\bigr\}$ is not dense in $\mathbb{R}$. This set coincides with $f(\mathcal{R}(M))$ for $f=\sum_{i=1}^n \lambda_i f_i$. Hence $\mathcal{R}(M)$ is not weakly dense in $X$, which is a contradiction.
This proves Proposition 2. Proposition 3. If $M$ is an all-round connected subset of a Banach space $X$, then the semigroup $\mathcal{R}(M)$ is weakly dense in $X$. Proof. Let $f \in X^\ast \setminus \{0\}$. Since the set $M$ is all-round, there exist $x, y \in M$ such that $f(x) < 0$ and $f(y) > 0$. It is clear that $f(M)$ is also connected, and therefore $f(M) \supset [f(x),f(y)]$. As a result, $f(\mathcal{R}(M)) \supset \mathcal{R}([f(x), f(y)])=\mathbb{R}$.
This proves Proposition 3. § 3. The semigroup generated by a subset of an $s$-ellipsoidThe next theorem on balancing vectors in an ellipsoid is the main ingredient in the proof of Theorem 1. Theorem B (Banaszczyk [4]). Let $\mathcal{E} \subset \mathbb{R}^d$ be an ellipsoid with centre at the origin and principal semiaxes $\lambda_1, \dots, \lambda_d$. Then for an arbitrary set of vectors ${v_1, \dots, v_n \in \mathcal{E}}$ there exist signs $\varepsilon_1, \dots, \varepsilon_n=\pm 1$ such that Given an orthonormal system $\{e_i\}_{i=1}^\infty$ in the space $H$ and a vector $\lambda\!=\!(\lambda_1, \lambda_2, \dots )$ with positive coordinates, consider the ellipsoid
$$
\begin{equation*}
\mathcal{E}_\lambda=\mathcal{E}_\lambda(\{e_i\}_{i=1}^\infty):=\biggl\{x=\sum_{i=1}^\infty x_i e_i\colon q_\lambda(x):=\sum_{i=1}^\infty \frac{x_i^2}{\lambda_i^2} \leqslant 1 \biggr\}
\end{equation*}
\notag
$$
with principal semiaxes $\lambda_1, \lambda_2, \dots$ . The $\ell_2$-norm of $\lambda$ will be denoted by $\| \lambda \|_2$. The next result is immediate from Theorem B. Corollary 1. Let $\mathcal{E}_\lambda \subset H$ be an ellipsoid, where $\lambda \in \ell_2$. Then for all $v_1, \dots, v_n \,{\in}\, \mathcal{E}_\lambda$ there exist signs $\varepsilon_1, \dots, \varepsilon_n=\pm 1$ such that Let $B(x,r)$ be the open ball in the Hilbert space $H$ of radius $r$ with centre at $x$. We need some auxiliary results. Lemma 1. Let $\mathcal{E}_\lambda \subset H$ be an ellipsoid, where $\lambda \in \ell_2$. Then for each $\varepsilon > 0$ there exist a positive number $\delta$ and an ellipsoid $\mathcal{E}_{\mu} \subset H$ such that $\mathcal{E}_\lambda \cap B(0, \delta) \subset \mathcal{E}_{\mu}$ and $\| \mu \|_2 \leqslant \varepsilon \| \lambda \|_2$. Proof. We can assume without loss of generality that $\varepsilon < 1$ and $\| \lambda \|_2=1$. Let $P_k$ be the orthogonal projection onto the subspace $\{x \in \ell_2\colon x_{k+1}=0, x_{k+2}=0, \dots\}$. Set
$$
\begin{equation*}
\mu:=\frac{\varepsilon}{2} P_k \lambda+2(\lambda - P_k\lambda),
\end{equation*}
\notag
$$
where $k$ is such that $\| \mu \|_2 \leqslant \varepsilon \| \lambda \|_2$. For $x \in \mathcal{E}_\lambda \cap B(0, \delta)$ we have
$$
\begin{equation*}
\begin{aligned} \, q_{\mu}(x)&=\frac{4}{\varepsilon^2} q_\lambda(P_k x)+\frac{1}{4}q_\lambda(x - P_k x) =\frac{1}{4} q_\lambda(x)+ \biggl(\frac{4}{\varepsilon^2} - \frac{1}{4}\biggr)q_\lambda(P_k x) \\ &\leqslant \frac{1}{4}+\biggl(\frac{4}{\varepsilon^2} - \frac{1}{4}\biggr)\max_{j=1, \dots, k} \frac{\delta^2}{\lambda_j^2} \to \frac{1}{4}, \qquad \delta \to 0. \end{aligned}
\end{equation*}
\notag
$$
Hence $x \in \mathcal{E}_{\mu}$ for sufficiently small $\delta$, that is, $\mathcal{E}_{\mu} \supset \mathcal{E}_\lambda \cap B(0, \delta)$.
This proves Lemma 1. Given $M \subset H$, consider the set
$$
\begin{equation*}
\operatorname{cone}(M, \mathbb{N})= \biggl\{\sum_{i=1}^n c_i x_i\colon c_i \geqslant 0,\,\sum_{i=1}^n c_i \in \mathbb{N},\, x_i \in M, \, n \in \mathbb{N} \biggr\}.
\end{equation*}
\notag
$$
Lemma 2. Let $M_1, \dots, M_\nu$ be subsets of the ellipsoids $\mathcal{E}_1, \dots, \mathcal{E}_\nu$, where ${\mathcal{E}_i\!=\!a_i\!+\!\mathcal{E}_\lambda}$, $a_i \in H$ and $\lambda \in \ell_2$. Then for each $x_i \in \operatorname{cone}(M_i, \mathbb{N})$ there exists $y_i \in \mathcal{R}(M_i)$ such that $\bigl\|\sum_{i=1}^\nu x_i - \sum_{i=1}^\nu y_i\bigr\| \leqslant 2\| \lambda \|_2$. Proof. Given $k \in \mathbb{N} \cup \{0\}$ and $M \subset H$, set
$$
\begin{equation*}
\mathcal{R}_k(M):=\biggl\{ \sum_{i=1}^m \frac{x_i}{2^k}\colon x_i \in M,\,\frac{m}{2^k} \in \mathbb{N}\biggr\}.
\end{equation*}
\notag
$$
Note that each $y \in \operatorname{cone}(M, \mathbb{N})$ can be approximated arbitrarily well by elements $y^{(k)} \in \mathcal{R}_k(M)$ for sufficiently large $k$. Indeed, by definition $y=\sum_{j=1}^m \alpha_j y_j$, where $\alpha_j > 0$, $y_j \in M$ and $\sum_{j=1}^m \alpha_j \in \mathbb{N}$. Let us construct in succession nonnegative integers $a_1^{(k)}, \dots, a_m^{(k)}$ such that
$$
\begin{equation}
0 \leqslant \sum_{j=1}^n \frac{a_j^{(k)}}{2^k} - \sum _{j=1}^n \alpha_j < \frac{1}{2^k}, \quad n =1, \dots, m-1,\quad\text{and} \quad \sum_{j=1}^m \frac{a_j^{(k)}}{2^k}=\sum _{j=1}^m \alpha_j.
\end{equation}
\tag{1}
$$
It is clear that $a_j^{(k)}/2^k \to \alpha_j$ as $k \to \infty$ for each $j=1, \dots, m$, and therefore
$$
\begin{equation*}
y^{(k)}:=\sum_{j=1}^m \frac{a_j^{(k)}}{2^k} y_j \to \sum_{j=1}^m \alpha_j y_j=y, \qquad k \to \infty.
\end{equation*}
\notag
$$
It remains to note that $y^{(k)} \in \mathcal{R}_k(M)$ in view of the last equality in (1). Let $\varepsilon > 0$ be arbitrary. We have $x_i \in \operatorname{cone}(M_i,\mathbb{N})$, and so for sufficiently large $N$ there exist $x_i^{(N)} \in \mathcal{R}_N(M_i)$, $i=1, \dots, \nu$, such that $\| x_i - x_i^{(N)}\| < \varepsilon/\nu$. We claim that for each $k=N, \dots, 1$ we can construct $x_i^{(k-1)} \in \mathcal{R}_{k-1}(M_i)$, $i=1, \dots, \nu$, such that $\| x^{(k)} - x^{(k-1)} \| \leqslant \| \lambda\|_2/{2^{k-1}}$, where $x^{(k)}=\sum_{i=1}^\nu x_i^{(k)}$. Assume that the elements $x_i^{(k)}=\sum_{j=1}^{2n_i}{y_{i, j}}/{2^k}$, $i=1, \dots, \nu$, have already been constructed. Each $v_{i, j}:=y_{i, 2j}-y_{i, 2j-1}$ lies in the ellipsoid $2\mathcal{E}_\lambda$, and so by Corollary 1 there exist signs $\varepsilon_{i, j}=\pm 1$ such that $\bigl\| \sum_{i=1}^\nu\sum_{j=1}^{n_i} \varepsilon_{i,j} v_{i,j}\bigr\| \leqslant 2\|\lambda\|_2.$ Let $\sigma_{i,j} \in \{2j-1, 2j\}$ be such that
$$
\begin{equation*}
\varepsilon_{i,j} v_{i, j}= y_{i, 2j-1}+y_{i, 2j} - 2y_{\sigma_{i,j}}.
\end{equation*}
\notag
$$
Also set
$$
\begin{equation*}
x_i^{(k-1)}:=\sum_{j=1}^{n_i} \frac{y_{\sigma_{i,j}}}{2^{k-1}} \in \mathcal{R}_{k-1}(M_i).
\end{equation*}
\notag
$$
Now the required estimate holds:
$$
\begin{equation*}
\| x^{(k)} - x^{(k-1)}\|=\biggl\| \sum_{i=1}^\nu\biggl( \sum_{j=1}^{2n_i} \frac{y_{i,j}}{2^k} - \sum_{j=1}^{n_i} \frac{y_{\sigma_{i,j}}}{2^{k-1}}\biggr)\biggr\|=\frac{1}{2^k}\biggl\| \sum_{i=1}^\nu \sum_{j=1}^{n_i} \varepsilon_{i,j} v_{i,j} \biggr\| \leqslant \frac{\| \lambda \|_2}{2^{k-1}}.
\end{equation*}
\notag
$$
It is clear that each $x_i^{(0)}$ lies in $\mathcal{R}(M)$. Hence for $x=\sum_{i=1}^\nu x_i$,
$$
\begin{equation*}
\begin{aligned} \, \operatorname{dist}(x,\mathcal{R}(M)) &\leqslant \| x - x^{(0)}\| \leqslant \| x - x^{(N)} \|+\sum_{k=1}^{N} \| x^{(k)} - x^{(k-1)} \| \\ &\leqslant \varepsilon+ \sum_{k=1}^{N}\frac{\| \lambda \|_2}{2^{k-1}} < \varepsilon+2\| \lambda \|_2 \to 2\| \lambda \|_2, \qquad \varepsilon \to 0, \end{aligned}
\end{equation*}
\notag
$$
which proves Lemma 2. Theorem 1. Let $M$ be a subset of a Hilbert space $H$ that lies in an $s$-ellipsoid $\mathcal{E}_\lambda$ with principal semiaxes $\lambda_1, \lambda_2, \dots$ . Then: (1) if $M$ is an all-round set in $H$, then $\overline{\mathcal{R}(M)}$ is an additive subgroup in $H$, and $\sup_{x \in H}\operatorname{dist}(x, \mathcal{R}(M)) \leqslant 2\bigl(\sum_{i=1}^\infty \lambda_i^2 \bigr)^{1/2}$; (2) $\overline{\mathcal{R}(M)}=H$ if and only if the additive semigroup $\mathcal{R}(M)$ is weakly dense in $H$. Proof. (1) We claim that $\overline{\mathcal{R}(M)}$ is an additive subgroup. The idea of the proof is to partition $M$ into sets $M_1, \dots, M_\nu$ lying in translations of a single ellipsoid ${\mathcal{E} \subset \mathcal{E}_\lambda \cap B(0, \delta)}$ with small sum of squared principal semiaxes, and then approximate the origin by an element $x \in \sum_{i=1}^\nu \operatorname{cone}(M_i, \mathbb{N})$. Next, by Lemma 2 we can approximate $x$ by an element $y \in \sum_{i=1}^\nu \mathcal{R}(M_i)$. The resulting point $y$, which is a sum of elements of some $(2\delta)$-net of the set $M$, approximates the origin, hence (since $\delta$ is arbitrary) it is clear that each element of $-M$ is approximated arbitrarily well by the additive subgroup $\mathcal{R}(M)$.
We assume without loss of generality that $M \subset B(0,1)$. Let $\varepsilon > 0$ be an arbitrary number. By Lemma 1 there exists $\delta \in (0,\varepsilon)$ such that $\mathcal{E}_{2\lambda} \cap B(0, \delta) \subset \mathcal{E}_{\mu}$, where $\| \mu \|_2 < \varepsilon/2$. Consider a finite cover of the set $M$ by the balls
$$
\begin{equation*}
B_1:=B(a_1, \delta),\quad \dots,\quad B_\nu:=B(a_\nu, \delta), \qquad a_1, \dots, a_\nu \in M.
\end{equation*}
\notag
$$
Note that for each $x \in \mathcal{E}_\lambda \cap B_i$,
$$
\begin{equation*}
x - a_i \in (\mathcal{E}_\lambda - \mathcal{E}_\lambda) \cap B(0,\delta)=\mathcal{E}_{2\lambda} \cap B(0, \delta) \subset \mathcal{E}_{\mu}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
M \subset \bigcup_{i=1}^\nu \mathcal{E}_\lambda \cap B_i \subset \bigcup_{i=1}^\nu (a_i+\mathcal{E}_\mu).
\end{equation*}
\notag
$$
Therefore, the set $M$ can be represented as the disjoint union
$$
\begin{equation*}
M=\bigsqcup_{i=1}^\nu M_i, \qquad M_i:=M \cap \biggl((a_i+\mathcal{E}_\mu) \setminus \bigcup_{j=1}^{i-1} (a_j+ \mathcal{E}_\mu)\biggr).
\end{equation*}
\notag
$$
We may assume that $M_i$ is nonempty. Let us now construct an element of small norm of the set $\sum_{i=1}^\nu \operatorname{cone}(M_i, \mathbb{N})$. Let $m_i \in M_i$ be arbitrary. The set $M$ is all-round, and so by Lemma 2.1 in [2] we have $\overline{\operatorname{cone}(M, \mathbb{N})}=H$. Therefore, there exist $u^{(n)} \in \operatorname{cone}(M,\mathbb{N})$ such that $u^{(n)} \to -\sum_{i=1}^\nu m_i$ as $n \to \infty$. It is clear that each $u^{(n)}$ is a sum of elements of the cones $\operatorname{cone}M_i$, that is,
$$
\begin{equation*}
u^{(n)}=\sum_{i=1}^\nu u^{(n)}_{i}, \qquad u^{(n)}_i \in \tau_i^{(n)} \operatorname{conv} M_i:=\{\tau_i^{(n)}u\colon u \in \operatorname{conv} M_i\},
\end{equation*}
\notag
$$
where $\tau_i^{(n)} \geqslant 0$ and $\sum_{i=1}^\nu \tau_i^{(n)} \in \mathbb{N}$ by $u^{(n)} \in \operatorname{cone}(M,\mathbb{N})$. Hence each $v^{(n)}:= \sum_{i=1}^\nu m_i+u^{(n)}$ tends to zero and is also a sum of elements of the cones $\operatorname{cone}M_i$, that is,
$$
\begin{equation*}
\begin{gathered} \, v^{(n)}=\sum_{i=1}^\nu v_i^{(n)}, \qquad v_i^{(n)}=m_i+ u^{(n)}_{i} \in t_i^{(n)} \operatorname{conv} M_i, \\ t_i^{(n)}=\tau_i^{(n)}+1 \geqslant 1, \qquad \sum_{i=1}^\nu t_i^{(n)} \in \mathbb{N}. \end{gathered}
\end{equation*}
\notag
$$
In what follows $[t]$ and $\{t\}$ denote the integer and fractional parts of the number $t \in \mathbb{R}$, respectively. We can assume without loss of generality that
$$
\begin{equation*}
\lim_{n \to \infty} \{t_i^{(n)}\}=t_i \in [0,1], \qquad i =1, \dots, \nu.
\end{equation*}
\notag
$$
By Dirichlet’s theorem on simultaneous approximation, we can find a natural number $N$ and nonnegative integers $n_i$ such that $|N t_i - n_i| < \varepsilon/\nu$ for each $i$. We choose a natural number $k$ so that $\bigl\| v^{(k)}\bigr\| < \varepsilon/N$ and $\bigl|\bigl\{t_i^{(k)}\bigr\} - t_i\bigr| < \varepsilon/(N \nu)$. The coefficients $k_i:=N[t_i^{(k)}]+n_i \in \mathbb{N}$ satisfy
$$
\begin{equation*}
\bigl|N t_i^{(k)} - k_i\bigr| \leqslant \bigl|N \{t_i^{(k)} \} - N t_i\bigr|+|N t_i - n_i| < \frac{2\varepsilon}{\nu}.
\end{equation*}
\notag
$$
Hence, for the elements $x_i^{(k)}:=v_i^{(k)}/t_i^{(k)} \in \operatorname{conv} M_i \subset B(0,1)$ we have
$$
\begin{equation*}
\begin{aligned} \, \biggl\| \sum_{i=1}^\nu k_i x_i^{(k)}\biggr\| &\leqslant \biggl\| \sum_{i=1}^\nu k_i x_i^{(k)} - N v^{(k)}\biggr\|+\| N v^{(k)}\| \\ & \leqslant \sum_{i=1}^\nu \biggl\| \bigl(k_i - N t_i^{(k)}\bigr) x_i^{(k)} \biggr\|+ \varepsilon < \nu \frac{2\varepsilon}{\nu}+\varepsilon=3\varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Next, $k_i x_i^{(k)} \in \operatorname{cone}(M_i, \mathbb{N})$ and $M_i \subset a_i+ \mathcal{E}_\mu$, and so by Lemma 2 there exist $y_i \in \mathcal{R}(M_i)$ such that $\bigl\| \sum_{i=1}^\nu k_i x_i^{(k)} - \sum_{i=1}^\nu y_i \bigr\| \leqslant 2\| \mu\|_2 < \varepsilon$. Hence
$$
\begin{equation}
y:=\sum_{i=1}^\nu y_i \in \mathcal{R}(M), \qquad \| y \| < 4\varepsilon.
\end{equation}
\tag{2}
$$
Now we claim that $-z \in \overline{\mathcal{R}(M)}$ for each $z \in M$. The element $z$ lies in some set $M_i$, $i \in \{1, \dots, \nu\}$. By (2) there exists $u_i \in M_i$ such that $y - u_i \in \mathcal{R}(M)$. Hence
$$
\begin{equation*}
\operatorname{dist}(-z, \mathcal{R}(M)) \leqslant \| -z - (y - u_i) \| \leqslant \| u_i - z\|+\| y\| < 2\delta+4\varepsilon < 6\varepsilon.
\end{equation*}
\notag
$$
Since $\varepsilon$ can be arbitrarily small, we have $-z \in \overline{\mathcal{R}(M)}$ for each $z \in M$, that is, $\overline{\mathcal{R}(M)}$ is an additive subgroup.
Let us estimate $\sup_{x \in H}\operatorname{dist}(x, \mathcal{R}(M))$. We have $\overline{\operatorname{cone}(M,\mathbb{N})}=H$, and now it is immediate from Lemma 2 (for $\nu=1$) that
$$
\begin{equation*}
\sup_{x \in H}\operatorname{dist}(x, \mathcal{R}(M)) = \sup_{x \in \operatorname{cone}(M,\mathbb{N})}\operatorname{dist}(x, \mathcal{R}(M)) \leqslant2\| \lambda\|_2.
\end{equation*}
\notag
$$
(2) It suffices to verify that the weak density of $\mathcal{R}(M)$ in $H$ implies the density of $\mathcal{R}(M)$ in $H$. We use the notation introduced in part (1) of the proof. We set
$$
\begin{equation*}
\operatorname{span}_0 (M_i)=\biggl\{\sum_{j=1}^n \lambda_j y_j\colon y_j \in M_i, \,\lambda_j \in \mathbb{R},\,\sum_{i=1}^n \lambda_j=0\biggr\}.
\end{equation*}
\notag
$$
Let us estimate the distance to $\mathcal{R}(M)$ of elements of the subspace
$$
\begin{equation*}
H_0:=\sum_{i=1}^\nu \operatorname{span}_0(M_i, \mathbb{N}).
\end{equation*}
\notag
$$
Since $\overline{\operatorname{span}_0(M_i)}$ is a subspace of $\overline{\operatorname{span}(M_i)}$ of codimension $0$ or $1$, it follows that $\overline{H_0}$ is a subspace of $H$ of codimension $\leqslant \nu$. Recall that $M_i \subset a_i+ \mathcal{E}_\mu$, where $\| \mu \|_2 < \varepsilon/2$. Hence by Lemma 2
$$
\begin{equation*}
\operatorname{dist}(u, \mathcal{R}(M)) \leqslant \varepsilon \quad \forall\, u \in U:= \sum_{i=1}^\nu \operatorname{cone}(M_i, \mathbb{N}).
\end{equation*}
\notag
$$
It is easily seen that $\operatorname{span}_0(M_i) \subset \operatorname{cone}(M_i, \mathbb{N}) - \operatorname{cone}(M_i, \mathbb{N})=\operatorname{cone}(\pm M_i, \mathbb{N})$, and therefore $H_0 \subset U - U$. Since $\overline{\mathcal{R}(M)}$ is an additive subgroup, for any element $x \in H_0$, $x=u' - u''$, $u', u'' \in U$, we have
$$
\begin{equation*}
\operatorname{dist}(x, \mathcal{R}(M)) \leqslant \operatorname{dist}(u', \mathcal{R}(M))+ \operatorname{dist}(-u'', \mathcal{R}(M)) \leqslant 2\sup_{u \in U}\operatorname{dist}(u, \mathcal{R}(M)) \leqslant 2\varepsilon.
\end{equation*}
\notag
$$
Now, for each $x \in H$ we estimate the distance $\operatorname{dist}(x, \mathcal{R}(M))$. Consider the orthogonal projections $P$ and $P^\perp=I - P$ onto the subspace $\overline{H_0}$, and let $H_0^\perp$ be the orthogonal complement of $\overline{H_0}$. By Proposition 1, the set $P^\perp(\mathcal{R}(M))$ is dense in $H_0^\perp$, that is, there exists a point $y \in \mathcal{R}(M)$ such that $\| P^\perp (x-y)\| < \varepsilon$. Next, for the element $P (x-y) \in H_0$ we find $y_0 \in \mathcal{R}(M)$ such that $\| P (x-y) - y_0\| \leqslant 2\varepsilon$. This implies that $\| P^\perp(y_0)\| \leqslant 2\varepsilon$, and so
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{dist}(x, \mathcal{R}(M))^2 \leqslant \| x - y - y_0\|^2=\| P(x-y-y_0) \|^2+\| P^\perp (x-y-y_0)\|^2 \\ &\quad\leqslant \| P(x-y) - y_0\|^2+2\| P^\perp(x-y)\|^2+2\| P^\perp (y_0)\|^2 \leqslant 4\varepsilon^2+2\varepsilon^2+8\varepsilon^2=14\varepsilon^2. \end{aligned}
\end{equation*}
\notag
$$
This estimate holds for each $\varepsilon > 0$, and therefore each $x \in H$ lies in $\overline{\mathcal{R}(M)}$.
This proves Theorem 1. Note that the estimate of the distance $\operatorname{dist}(x, \mathcal{R}(M))$ in part (1) of Theorem 1 is optimal up to a constant. Remark 1. Let $\{e_i\}_{i=1}^\infty$ be an orthonormal basis for $H$. Then for each ellipsoid $\mathcal{E}_\lambda=\mathcal{E}_\lambda(\{e_i\}_{i=1}^\infty)$ the set $M= \{\pm\lambda_i e_i\}_{i=1}^\infty \subset \mathcal{E}_\lambda$ is all-round in the space $H$ and $\sup_{x \in H}\operatorname{dist}(x, \mathcal{R}(M))=\bigl(\sum_{i=1}^\infty |\lambda_i|^2\bigr)^{1/2}/2$. That $M$ is all-round in $H$ is clear. Note that for each $x=\sum_{i=1}^\infty x_i e_i \in H$,
$$
\begin{equation*}
\operatorname{dist}(x,R(M))^2=\sum_{i=1}^\infty \min_{k \in \mathbb{Z}}|x_i - k \lambda_i|^2 \leqslant \sum_{i=1}^\infty \frac{\lambda_i^2}{4}.
\end{equation*}
\notag
$$
On the other hand, for $x_n:=(1/2) \sum_{i=1}^n \lambda_i e_i$ we have
$$
\begin{equation*}
\operatorname{dist}(x_n, \mathcal{R}(M))^2=\sum_{i=1}^n \frac{\lambda_i^2}{4} \to \sum_{i=1}^\infty \frac{\lambda_i^2}{4} , \qquad n \to \infty.
\end{equation*}
\notag
$$
A direct consequence of Theorem 1 is the following test for the density of an additive semigroup generated by a subset of an $s$-ellipsoid. Corollary 2. Let $M \subset H$ be a subset of an $s$-ellipsoid. Then the additive semigroup $\mathcal{R}(M)$ is dense in $H$ if and only if $M$ is an all-round set and there is no nonzero $z \in H$ such that $\langle M, z\rangle \subset \mathbb{Z}$. § 4. Application to Lipschitz and holomorphic mappingsTo verify that a set lies in an $s$-ellipsoid can be a nontrivial problem. To this end the following criterion can be used. Theorem C (Sudakov [5]). A subset $M$ of a Hilbert space $H$ lies in an $s$-ellipsoid if and only if for each orthonormal basis $\{e_k\}_{k=1}^\infty$ of $H$. Proposition 4. Let $K \subset \mathbb{R}$ be a compact set and $f\colon K \to H$ be a Lipschitz map. Then the image $f(K)$ lies in some $s$-ellipsoid. Proof. We may assume without loss of generality that $K \subset [0,1]$. By Kirszbraun’s theorem the Lipschitz mapping $f\colon K \to H$ extends to a map $f\colon [0,1] \to H$ with the same Lipschitz constant. We claim that $f([0,1])$ lies in some $s$-ellipsoid. Let $\gamma\colon [0,1] \to H$ be a rectifiable curve which we must place in a $s$-ellipsoid $\mathcal{E}$. We can assume without loss of generality that $\gamma$ has length $|\gamma|=1$ and $\gamma(0)=0$. Let $\{e_j\}_{j=1}^\infty$ be an arbitrary orthonormal basis of $H$. Set $g_j(\gamma):=\sup_{t \in [0,1]}|\langle\gamma(t),e_j\rangle|$. Let us find $t_j \in [0,1]$ such that $|\langle \gamma(t_j), e_j\rangle|=g_j(\gamma)$. We claim that $\sum_{j=1}^n g_j(\gamma)^2 \leqslant 1$ for each $n \in \mathbb{N}$. We can assume that $t_1 \leqslant t_2 \leqslant \dots \leqslant t_n$ (this can always be achieved by relabelling the basis vectors $e_1, \dots, e_n$). For each $x=\sum_{j=1}^\infty x_j e_j \in H$ consider the vector $x_+:=\sum_{j=1}^\infty |x_j| e_j$. By definition $\| x_+\|=\| x \|$. We set $v:=\sum_{j=1}^n \bigl( \gamma(t_j) - \gamma(t_{j-1}) \bigr)_+$ (here $t_0=0$). It is easily seen that $g_j(\gamma) \leqslant \langle v, e_j\rangle$ for all $j=1, \dots, n$ and $\| v \| \leqslant \sum_{j=1}^n \| \gamma(t_j) - \gamma(t_{j-1})\| \leqslant |\gamma|=1$. Hence $\sum_{j=1}^n g_j(\gamma)^2 \leqslant \| v \|^2 \leqslant 1$. Letting $n\to\infty$, we find that $\sum_{j=1}^\infty g_j(\gamma)^2 \leqslant 1$, and so by Theorem C the curve $\gamma$ is contained in an ellipsoid with sum of squared semiaxes at most 1.
This proves Proposition 4. It is unknown whether or not Proposition 4 can be extended to the case of a compact set $K \subset \mathbb{R}^d$ for $d=2$; however, for $d \geqslant 3$ this is not true. On the other hand this extension is possible if the Lipschitz condition is replaced by the condition that $f$ is holomorphic. The proof of this fact is based on an estimate for Kolmogorov widths of the image of a compact set under a holomorphic map. Recall that the Kolmogorov width of a subset $M$ of a Banach space $X$ is defined by $d_n(K)=\inf_{L_n} \sup_{x \in K} \operatorname{dist}(x,L_n)$, where the infimum is taken over all $n$-dimensional subspaces $L_n$ of $X$. Theorem D (Cohen and DeVore [6]). Let $X$ and $ Y$ be Banach spaces over $\mathbb{C}$, $F$ be a holomorphic mapping from an open set $U \subset X$ to $Y$, $\sup_{x \in U}\| F(x)\|_Y < \infty$, and let $K \subset U$ be a compact set. Then for all $s > 1$ and $t < s - 1$, Lemma 3. Let $K$ be a compact subset of a complex Hilbert space $H^{\mathbb{C}}$, and let $\sup_{n \geqslant 1}n^s d_n(K) < \infty$ for some $s > 1$. Then there exists an ellipsoid containing $K$ with finite sum of squared semiaxes, where $\{e_j\}_{j=1}^\infty$ is an orthonormal system. Proof. Let $C \geqslant \operatorname{diam}K$ be such that $\sup_{n \geqslant 1}n^s d_n(K) < C$. For each nonnegative integer $k$, let $\widetilde{L}_k \subset H$ be a $2^k$-dimensional subspace such that $\sup_{z \in K}\operatorname{dist}(z, \widetilde{L}_k) < C 2^{-sk}$. We set $L_{k}:=\widetilde{L}_0+\dots+\widetilde{L}_{k-1}$, $k \in \mathbb{N}$ and $C':=2^s C$. It is clear that $L_k$ is a nested sequence of subspaces,
$$
\begin{equation*}
\sup_{z \in K}\operatorname{dist}(z, L_k) < C 2^{-s(k-1)}=C' 2^{-sk} \quad\text{and} \quad \dim L_k \leqslant \sum_{j=0}^{k-1} 2^j=2^{k}-1.
\end{equation*}
\notag
$$
Let $\{e_j\}_{j=1}^\infty$ be an orthonormal system such that $L_k = \operatorname{span}\{e_1, \dots, e_{n_k}\}$ for each $k$. Then for all $z=\sum_{j=1}^\infty z_j e_j \in K$ and $m \in 2^{k}, \dots, 2^{k+1}-1$,
$$
\begin{equation}
|z_m| \leqslant \biggl(\sum_{j=2^k}^{\infty}|z_j|^2\biggr)^{1/2} \leqslant \operatorname{dist}(z,L_k) \leqslant C' 2^{-ks}.
\end{equation}
\tag{4}
$$
We set $\gamma:=\bigl(C'\sum_{k=0}^\infty 2^{-k(s-1)}\bigr)^{1/2}$ and consider the vector $\lambda=(\lambda_1, \lambda_2, \dots) \,{\in}\, \ell_2$ with coordinates $\lambda_m:=\gamma 2^{-sk/2}$, where $m \in 2^k, \dots, 2^{k+1}-1$ and $k \in \mathbb{N} \cup \{0\}$. It is clear that the sum of squared semiaxes of the ellipsoid $\mathcal{E}_\lambda^{\mathbb{C}}$ (see (3)) is finite. Now we show that each element $z \in K$ lies in $\mathcal{E}_\lambda^{\mathbb{C}}$. Indeed, by (4)
This proves Lemma 3. The next result follows easily from Theorem D and Lemma 3. Corollary 3. Let $K \subset \mathbb{C}^d$ be a compact set and $f\colon K \to H^\mathbb{C}$ be a holomorphic map. Then $f(K)$ lies in an $s$-ellipsoid of the realified space $H^\mathbb{C}$. In addition, if the additive semigroup $\mathcal{R}(f(K))$ is weakly dense in the realified space $H^\mathbb{C}$, then it is dense in $H^\mathbb{C}$. Proof. By the assumptions of the corollary $K \subset \mathbb{C}^d$, and so $d_n(K)=0$ for $n \geqslant d$. Therefore, $\sup_{n \geqslant 1}n^s d_n(K) < \infty$ for each $s > 1$. Hence by Lemma 3 the image $f(K)$ lies in the ellipsoid $\mathcal{E}_\lambda^\mathbb{C}$, $\lambda \in \ell_2$. It remains to note that in the realification of $H^\mathbb{C}$ this ellipsoid has principal semiaxes $\lambda_1, \lambda_1, \lambda_2, \lambda_2, \dots$, and so it is an $s$-ellipsoid. If the additive semigroup $\mathcal{R}(f(K))$ is weakly dense in the realified $H^\mathbb{C}$, then it is dense in $H^\mathbb{C}$ by part (2) of Theorem 1.
This proves Corollary 3. To conclude, we derive from Corollary 3 a generalization of Korevaar’s theorem on approximation of holomorphic functions in a bounded simply connected domain $D \subset \mathbb{C}$ by simple partial fractions $\sum_{i=1}^n {1}/(z - w_i)$ with poles $w_i \in \partial D$ (see [7] and [2], § 3.1). Let $\mathrm{AC}(K)$ be the space of functions continuous on the compact set $K$ and holomorphic in the interior of $K$; this space is equipped with the Chebyshev norm. Theorem 2. Let $D$ be a bounded simply connected domain in $\mathbb{C}$, $F \subset \mathbb{C}$ be a compact set and $f \not\equiv 0$ be a holomorphic function in $\overline{\mathbb{C}} \setminus F$ such that $f(\infty)=0$. Then the additive semigroup $\mathcal{R}(M)$ generated by the set $M= \{f(z-w)\colon w \in \partial D\}$ is dense in the space $\mathrm{AC}(K)$ for each compact set $K$ with connected complement such that $K - F \subset D$. Proof. In what follows the $\delta$-neighbourhood of a set $V \subset \mathbb{C}$ is denoted by $V_\delta:= \bigcup_{v \in V} B(v,\delta)$. Since $K - F$ is a compact set, we have
$$
\begin{equation*}
\inf_{z \in K - F} \operatorname{dist}(z, \mathbb{C} \setminus D)=: 3\delta > 0.
\end{equation*}
\notag
$$
Consider an open neighbourhood $U \subset (\partial D)_\delta$ of the boundary $\partial D$, and consider a smooth Jordan contour $S$ such that $K \subset \operatorname{Int}S \subset K_\delta$, where $\operatorname{Int}S$ is the bounded connected component of the set $\mathbb{C} \setminus S$. We have $\overline{\operatorname{Int} S} - F_\delta \subset D \setminus U$; this is equivalent to saying that
$$
\begin{equation}
\overline{\operatorname{Int}S} - (\mathbb{C} \setminus (D \setminus U)) \subset \mathbb{C} \setminus F_\delta
\end{equation}
\tag{5}
$$
(it is easily seen that the inclusions $X - Y \subset Z$ and $X - (\mathbb{C} \setminus Z) \subset (\mathbb{C} \setminus Y)$ are equivalent for all sets $X, Y, Z \subset \mathbb{C}$). It is immediate from inclusion (5) that each function $f(z - u)$, $u \in \mathbb{C} \setminus (D \setminus U)$ is holomorphic in the $\delta$-neighbourhood of the compact set $\overline{\operatorname{Int} S}$. First we claim that $\mathcal{R}(M)$ is dense in the Smirnov space $\mathrm{AL}_2(S)$ (this is the completion of the space $\mathrm{AC}(\overline{\operatorname{Int}S})$ with respect to the norm of $L_2(S, |dz|)$). Consider the map
$$
\begin{equation*}
\mathscr{F}\colon U \to \mathrm{AL}_2(S), \qquad u \mapsto f(\cdot - u).
\end{equation*}
\notag
$$
It is clear that $U \subset \mathbb{C} \setminus (D \setminus U)$, and so (5) implies that $S - U \subset \mathbb{C} \setminus F_\delta$. Hence $\mathscr{F}$ is holomorphic and satisfies the hypotheses of Corollary 3. Consequently, ${M=\mathscr{F}(\partial D)}$ is contained in the $s$-ellipsoid of the realified space $\mathrm{AL}_2(S)$. In addition, $M$ is connected, and so by Proposition 3, for a proof that $\mathcal{R}(M)$ is dense in $\mathrm{AL}_2(S)$ it suffices to verify that $\mathcal{R}(M)$ is an all-round set in this space. If this were not so, then there would exist a nonzero element $h \in \mathrm{AL}_2(S)$ such that
$$
\begin{equation}
\langle f(\,\cdot - \zeta), h \rangle=\operatorname{Re} \int_S f(z-\zeta) \overline{h(z)}\,|dz|\geqslant0 \quad \forall\, \zeta \in \partial D.
\end{equation}
\tag{6}
$$
Now we claim that $\langle f(\cdot - \zeta), h\rangle=0$ for each $\zeta \in \partial D$. From (5) and since $f(\infty)=0$, it is clear that the function is holomorphic in $(\mathbb{C} \setminus \overline{D})_\delta$ and $\lim_{\zeta \to \infty} H(\zeta)=0$. Hence $H(z)$ is holomorphic in the domain $(\overline{\mathbb{C}} \setminus \overline{D})_\delta$. Consequently, $\operatorname{Re} H(\zeta)$ is a harmonic function in the domain $\Omega:=\mathbb{C} \setminus \overline{D}$ and $\operatorname{Re} H(\infty)=0$. Now we see from (6) and the maximum principle that $\operatorname{Re} H(z) \geqslant 0$ in $\Omega$. Therefore, the function $\operatorname{Re} H(1/\zeta)$, which is harmonic in a neighbourhood of the origin, has a local minimum at the origin. Hence this function is zero in this neighbourhood by the maximum principle, that is, $\operatorname{Re} H(\zeta)=0$ near $\infty$, and therefore this function vanishes in the whole domain $\Omega$ and on its boundary $\partial D$. As a result, Next, by Corollary 6 in [1] the semigroup $\mathcal{R}(\pm M)$ is dense in $\mathrm{AC}(\overline{\operatorname{Int}S})$ (as the set $E$ in Corollary 6 we can take $E=\partial D$), and therefore in $\mathrm{AL}_2(S)$. Hence there exists a sequence of functions $f_n \in \mathcal{R}(\pm M)$ converging to $h$. Therefore, $\| h \|_{\mathrm{AL}_2(S)}^2=\lim_{n \to \infty} \langle f_n, h\rangle=0$ by (7), that is, $h=0$, which contradicts the choice of this element. Thus, the semigroup $\mathcal{R}(M)$ is dense in $\mathrm{AL}_2(S)$, and so each polynomial is approximated arbitrarily well in $\mathrm{AL}_2(S)$ by elements of $\mathcal{R}(M)$. Now by Cauchy’s integral formula this polynomial can be approximated arbitrarily well by elements of $\mathcal{R}(M)$ uniformly on $K$. Hence these polynomials are dense in $\mathrm{AC}(K)$ by Mergelyan’s theorem.
This proves the theorem. The author is grateful to P. A. Borodin and to the referee for valuable suggestions and also to Yu. A. Skvortsov for drawing our attention to Sudakov’s paper. 
Citation in format AMSBIB
\Bibitem{Shk25} |
|
||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|