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On universal (in the sense of signs) Fourier series with respect to the Walsh system M. G. Grigoryan Yerevan State University, Yerevan, Republic of Armenia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 6, DOI: https://doi.org/10.4213/sm10014 
Bibliographic databases:
 § 1. IntroductionWe consider the problem of the existence of (universal) functions whose Fourier–Walsh series are universal in the class of almost everywhere finite measurable functions in the sense of signs. Existence of functions and series which are universal in some or other sense has extensively been studied in the theory of functions of a real or a complex variable. First examples of universal functions were constructed by Birkhoff [1] in the complex analysis setting (every entire function was shown to be representable in any disc by uniformly convergent translations of the universal function) and by Marcinkiewicz [2] in the real analysis setting (any measurable function was shown to be representable as the limit almost everywhere of some sequence of difference relations of a universal function (see also [3]–[6]). Recently, this author [7]–[14] obtained some results on the existence and description of the structure of functions (universal functions) whose Fourier series with respect to a given classical system are universal (in one sense or another, for various function classes). The concept of universal series dates back to Men’shov [16] and Talalyan [17]. The most general results in this direction were obtained by Men’shov, Talalyan and their students (see [17]–[24]). We need the following notation. Let $L^{0}[0,1]$ be the class of measurable functions which are almost everywhere finite on $[0,1]$ and $M[0,1]$ be the class of all measurable functions on $[0,1]$. A sequence $\{f_k(x)\}_{k=1}^{\infty}\subset L^{0}[0,1]$ is said to converge to $f(x)$ in $L^{0}[0,1]$ (respectively, in $M[0,1]$) if $\{f_k(x)\}_{k=1}^{\infty}$ converges to $f(x)$ almost everywhere on $[0,1]$ (respectively, almost everywhere or in measure on $[0,1]$). Let $E\subseteq[0,1]$ be a measurable set, $|E|$ be the Lebesgue measure of $E\subseteq [0,1]$, and let $L^{p}(E) $ be the class of all measurable functions on $E$ such that $\displaystyle\int_{E}|f(x)|^{p}\,dx<\infty$, $p>0$. Let $f,f_k\in L^{p}[0,1]$, $k\in\mathbb{N}$ ($\mathbb{N}$ denotes the set of natural numbers). A sequence $\{{f}_k(x)\}_{k=1}^{\infty}$ is said to converge to $ f(x)$ in $L^{p}[0,1]$ if $\{f_k(x)\}_{k=1}^{\infty}$ converges to $ f(x)$ in $L^{p}[0,1]$, that is,
$$
\begin{equation*}
\lim_{k\to\infty}\int_0^{1}|{f}_k(x)-f(x)|^{p}\,dx=0.
\end{equation*}
\notag
$$
A series $\sum_{k=1}^{\infty}{f}_k(x)$, ${f}_k\in L^{p}[0,1]$, $p\geqslant0$, is said to be universal in $L^{p}[0,1]$ (respectively, in $M[0,1]$) if for each $f\in L^{p}[0,1]$ (respectively $f\in M[0,1]$) there exists an increasing subsequence of natural numbers $n_k$ such that the subsequence of partial sums with indices $n_k$ of the series $\sum_{k=1}^{\infty}{f}_k(x)$ converges to $f(x)$ in $L^{p}[0,1]$ (respectively, in $M[0,1]$). Let $\Phi:=\{\varphi_k(x)\}_{k=0}^{\infty}$ be an $L^{2}[0,1]$-complete orthonormal system of bounded functions, and, given a function $f\in L^{1}[0,1]$, let
$$
\begin{equation}
c_k(f):=\int_0^{1}f(x)\varphi_k(x)\,dx, \qquad k\in\mathbb{N} \cup\{0\},
\end{equation}
\tag{1.1}
$$
be the Fourier coefficients of $f$, and let
$$
\begin{equation}
S_{m}(f): =\sum_{k=0}^{m}c_k(f)\varphi_k(x), \qquad m\in\mathbb{N}\cup\{0\},
\end{equation}
\tag{1.2}
$$
be the partial sums of the Fourier series $\sum_{k=0}^{\infty}c_k(f)\varphi_k(x)$ of $f$ in the system $\{\varphi_k(x)\}_{k=0}^{\infty}$. Let $S$ be any of the spaces $L^{p}[0,1]$, $p\in({0,1})$, $L^{0}[0,1]$, and $M[0,1]$. The cardinality of a finite set $\Omega$ is denoted by $\#(\Omega) $. The following definition is required for the formulation of some of our results. Definition 1. Let $\Omega\subset \Lambda\subseteq\mathbb{N}$. The density of $\Omega$ with respect to $\Lambda$ is defined by Definition 2. Given a class $S$, we say that, with respect to a system $\{\varphi_k(x)\}_{k=0}^{\infty}$, a function $U\in L^{1}[a,b]$ is (1) universal for $S$ if the Fourier series of $U(x)$ in this system is universal for $S$, (2) conditionally universal for $S$ if there exists a sequence of signs $\{\delta_k= \pm1\}_{k=0}^{\infty}$ such that the series $\sum_{k=0}^{\infty}\delta_kc_k(U)\varphi_k(x)$ is universal for $S$, (3) almost universal for $S$ if there exists a sequence of signs $\{\delta_k= \pm1\}_{k=0}^{\infty}$ with $\rho(\Omega)_{\Lambda}=1$ (where $\Omega(U)= \{k\in\Lambda(U)=\operatorname{spec}(U): \,\delta_k=1\}$) such that the series $\sum_{k=0}^{\infty}\delta_kc_k(U)\varphi_k(x)$ is universal for $S$, (4) universal in the sense of signs for $S$ if for each function $f \in S$ there exists a sequence of signs $\{\delta_k\,{=}\,\pm1\}_{k=0}^{\infty}$ such that the series $\sum_{k=0}^{\infty}\delta_kc_k(U)\varphi_k(x)$ converges to $f(x)$ in $S$, (5) universal in the sense of permutations for $S$ if the Fourier series of $U(x)$ is universal for $S$ in the sense of permutations, that is, for each function $f\in S$ the series $\sum_{k=0}^{\infty}c_k(U)\varphi_k(x)$ can be permuted so that the resulting series $\sum_{k=1}^{\infty}c_{\sigma(k)}(U)\varphi_{\sigma (k)}(x)$ converges to $f(x)$ in $S$. Definition 3. We say that a function $U\in L^{1}[0,1]$, a measurable set $E\subset [0,1]$, and a sequence of signs $\mathbf{\delta}=\{\delta_k=\pm1\}_{k=0}^{\infty}$ form a universal triple ($U,E,\mathbf{\delta}$) in the sense of modification for a class $S$ with respect to a system $ \Phi:=\{\varphi_k(x)\}_{k=0}^{\infty}$ if Note that it follows from Kolmogorov’s theorem (see [26]), which asserts that the trigonometric Fourier series of each integrable function is $L^{p}$-convergent for $p\in(0,1)$, that there exists no integrable function whose trigonometric Fourier series is universal for the class $M[0, 2\pi]$ of all measurable functions. In the same way it follows from Watari’s theorem (see [25]), which asserts that the Fourier–Walsh series of each integrable function is $L^{p}$-convergent for $p\in(0,1)$, that there exists no integrable function whose Fourier–Walsh series is universal for the class $M[0, 2\pi]$ of all measurable functions. Hence there exists no function universal for the class $L^{p}[0,1]$, $p\in[0,1)$, with respect to the trigonometric system (or with respect to the Walsh system). We also note that there exists no function universal for the class $L^{p}[0,1]$, $p\in(0,1)$, with respect to the Vilenkin, Haar or Franklin systems. Nevertheless, in [8]–[10] it was shown that, for the classes $L^{p}$, $p\in (0,1)$, there exist conditionally universal functions with respect to the Walsh and trigonometric systems alike. We also note that in [10] we constructed a universal triple ($U,E,\mathbf{\delta}$) in the sense of modification for the classes $L^{p}[0,1]$, $p\in(0,1)$, with respect to the Walsh system. Moreover, the following result holds. Theorem 1. There exists an integrable function $U$ with Fourier–Walsh series convergent everywhere on $[0,1)$ and in $L^{1}[0,1)$ such that: (1) $U$ is an almost universal function for the class $L^{p}[0,1]$, $p\in(0,1)$, with respect to the Walsh system; (2) for any $\varepsilon>0$ there exist a measurable set $E\subset [0,1]$, $|E|>1-\varepsilon$, such that for each $f\in L^{1}[0,1]$ there exists a function $\widetilde{f}\in L^{1}[0,1]$ such that $\widetilde{f}(x)=f(x)$ on $E$ and $ |c_k(\widetilde{f})|=|c_k(U)|$, $k=0,1,2,\dots$ . The papers [8] and [9] were concerned with the existence of functions universal in the sense of signs for the classes $L^{p}$, where $p\in(0,1)$, with respect to the Walsh system (trigonometric system, respectively). The following theorem was proved in [14]. Theorem 2. There exist an integrable function $U\in L^{1}[0,1]$ with $L^{1}[0,1]$-convergent Fourier–Walsh series with monotonically decreasing coefficients, and there exist natural numbers $\{N_{m}\}_{m=1}^{\infty}$ such that: (1) for each function $f\in M[0,1]$ there exists a sequence of signs $\{\delta_k=\pm1\}_{k=0}^{\infty}$ such that the subsequence $\sum_{k=0}^{N_{m}}\delta_kc_k(U)W_k(x)$ converges to $f(x)$ almost everywhere on $[0,1]$; (2) the function $U$ is universal for the class $M[0,1]$ with respect to the Walsh system in the sense of signs in the case of convergence in measure (that is, for each function $f\in S$ there exists a sequence of signs $\{\delta_k=\pm1\}_{k=0}^{\infty}$ such that the series $\sum_{k=1}^{\infty}\delta_kc_k(U)W_k(x)$ converges to $ f(x)$ in measure on $[0,1]$). Remark 1. Theorem 2 is sharp in the following sense: in this theorem one cannot replace $\{N_{m}\}_{m=1}^{\infty}$ by $m$, because it is known (see [27]) that a Walsh series cannot converge to $\infty$ on a set of positive measure. Consequently, there does not exist a function which is universal in the sense of signs with respect to the Walsh system for the class $M[0,1]$ in the case of convergence almost everywhere. However, there exists a function $U\in L^{1}[0,1]$ universal in the sense of signs with respect to the Walsh system for the class $M[0,1]$ in the case of convergence in measure (see [14]), and, in addition, one can construct a function universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1]$ in the case of convergence almost everywhere. In the present paper we prove the following theorem, which was announced in [13]. Theorem 3. There exists a function $U\in L^{1}[0,1]$ whose Fourier–Walsh series has monotone decreasing coefficients and converges in $L^{1}[0,1]$ and almost everywhere on $[0,1]$ such that $U$ is universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1]$. Remark 2. The author does not know whether Theorems 1–3 hold for the trigonometric system. However, these theorems do not hold for general orthonormal systems; in particular, the conclusion of Theorem 3 is not true for the system $\{f_{n}(x)\}$ constructed by Kashin in [28] (he constructed an $L^{2}[0,1]$-complete orthonormal system $\{f_{n}(x)\}$ of bounded functions such that if a series $\sum_{k=1}^{\infty}a_kf_k(x)$ converges almost everywhere on $[0,1]$, then $\sum_{k=1}^{\infty}a_k^{2} < \infty$), that is, there does not exist a function $U\in L^{1}[0,1]$ universal in the sense of signs with respect to the system $\{f_{n}(x)\}$ for the class $L^{0}[0,1]$. It is also worth pointing out that for all $p\geqslant1 $ and a bounded orthonormal system $\{\varphi_{n}(x)\}$ there exists no function $U\in L^{1}[0,1]$ which is universal in the sense of signs with respect to the system $\{\varphi_{n}(x)\}$ for the class $L^{1}[0,1]$. Indeed, if for some $p\geqslant1$ there existed a function $U\in L^{1}[0,1]$ which is universal in the sense of signs with respect to a bounded orthonormal system $\{\varphi_{n}(x)\}$ for the class $L^{p}[0,1]$, $p\geqslant1$, then for any function $ g(x)\in L^{p}[0,1]$, $p\geqslant1$, $c_1(g)\neq0$, there would exist numbers $\{\delta_k=\pm 1\}_{k=0}^{\infty}$ and $\{\varepsilon_k=\pm 1\}_{k=0}^{\infty}$ such that
$$
\begin{equation*}
\lim_{m\to\infty}\int_0^{1} \biggl|\sum_{k=0}^{m}\delta_kc_k(U)\varphi_k(x)- g(x)\biggr| \, dx=0
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{m\to\infty}\int_0^{1} \biggl| \sum_{k=0}^{m}\varepsilon_kc_k(U)\varphi_k(x)- 4g(x)\biggr|\, dx =0.
\end{equation*}
\notag
$$
Since $\delta_1c_1(U)=c_1(g)$ and $\varepsilon _1c_1(U)=c_1(4g)=4c_1(g)$, this immediately gives $\varepsilon_1=4\delta_1$, which is a contradiction. A similar analysis shows that there does not exist a function which is conditionally universal (and, therefore, almost universal) with respect to the Walsh system for the class $L^{1}[0,1]$. Remark 3. It is worth pointing out that the existence of universal functions depends (as our results show) on the type of universality, the system under consideration, the convergence in question and the space, and so the problem here is quite extensive. We also note that any measurable, almost everywhere finite function can be transformed into a universal function in the sense of signs with respect to the Walsh system (in particular, for the class $L^{0}[0,1]$) by changing its values on a set of arbitrarily small measure. The following stronger result holds. Theorem 4. There exists a function $U\in L^{1}[0,1]$ with $L^{1}[0,1]$-convergent Fourier–Walsh series with monotone decreasing coefficients such that: (1) $U$ is universal in the sense of signs with respect to the Walsh system for the class $L^{0} [0,1]$ in the case of convergence almost everywhere and universal in the sense of signs with respect to the Walsh system for the class $M[0,1]$ in the case of convergence in measure; (2) for any $\varepsilon>0$ there exists a measurable set $E\subset [0,1]$, $|E|>1-\varepsilon$, such that, for each function $f\in L^{1}[0,1]$ there exists a function $\widetilde{f}\in L^{1}[0,1]$ such that $\widetilde{f}(x)=f(x)$ on $E$ and $|c_k(\widetilde{f})|=|c_k(U)|$, $k=0,1,2,\dots$ . The next result is a corollary to Theorem 4. Theorem 5. For each $\varepsilon>0$ there exist a measurable set $E\subset [0,1]$, $|E|>1-\varepsilon$, and, for each function $f\in L^{1}[0,1]$, a function $\widetilde{f}\in L^{1}[0,1]$ such that $\widetilde{f}(x)=f(x)$ on $E$ and $\widetilde{f}(x)$ is universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1]$ in the case of convergence almost everywhere and universal in the sense of signs with respect to the Walsh system for the class $M[0,1]$ in the case of convergence in measure. The author will present the proofs of Theorems 1 and 4 elsewhere. Thus, we have the following picture: (1) there does not exist a function which is universal with respect to the classical systems for the class $M[0,1]$ (and, therefore, for the classes $L^{p}[0,1]$, $p\in [0,1)$) in the case of convergence almost everywhere; (2) there exists an almost universal function with respect to the Walsh system for the class $L^{p}[0,1]$, $p\in (0,1)$ (and therefore for the classes $L^{0}[0,1]$ and $M[0,1]$); (3) there exists a function $U\in L^{1}[0,1]$ which is universal in the sense of signs with respect to the Walsh system for the class $L^{p}[0,1]$, $p\in(0,1)$; (4) there exists a function $U\in L^{1}[0,1]$ which is universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1]$ (in the case of convergence almost everywhere); (5) there exists a function $U$ which is universal in the sense of signs with respect to the Walsh system for the class $M[0,1]$ in the case of convergence in measure, but no function universal in the sense of signs for the class $M[0,1]$ in the case of convergence almost everywhere; (6) there does not exist a function universal in the sense of signs with respect to the Walsh system for the class $L^{1}[0,1]$, but there exists an asymptotically universal function in the sense of signs (see [10]), that is, there exist a function $U\in L^{1}[0,1]$ and measurable sets $E_{n}\subset E_{n+1}\subset[0,1]$, $n=1,2,\dots$, $\lim_{n\to\infty}|E_{n}|=1$, such that for any function $f\in L^{1}[0,1]$ there exists a sequence of signs $\{\varepsilon_k=\pm1\}_{k=0}^{\infty}$ such that for each $n\in\mathbb{N}$
$$
\begin{equation*}
\lim_{m\to\infty}\int_{E_{n}} \biggl| \sum_{k=0}^{m}\varepsilon_kc_k(U)W_k(x)- f(x)\biggr| \, dx=0;
\end{equation*}
\notag
$$
(7) there does not exist a conditionally universal function with respect to the Walsh system for the class $ L^{1}[0,1]$, but there exists an asymptotically almost universal function, that is, there exist a function $U\in L^{1}[0,1]$, measurable sets $E_{n}\subset E_{n+1}\subset[0,1]$, $n=1,2,\dots$, $ \lim_{n\to\infty}|E_{n}|=1$, and a sequence of signs $\{\delta_k=\pm1\}_{k=0}^{\infty }$, $\rho(\Omega)_{\Lambda}=1$ (where $\Omega(U)= \{k\in\Lambda(U)=\operatorname{spec}(U),\delta_k=1\}$) such that for any function $f\in L^{1}[0,1]$ there exists a subsequence of natural numbers $\{N_{m}\}_{m=1}^{\infty}\nearrow $ such that for each $n\in\mathbb{N}$,
$$
\begin{equation*}
\lim_{m\to\infty}\int_{E_{n}} \biggl| \sum_{k=0}^{N_{m}}\delta_kc_k(U)W_k(x)- f(x)\biggr| \, dx=0.
\end{equation*}
\notag
$$
However, the author does not know an answer to the following related questions. Question 1. Does there exist a function $ U\in L^{1}[0,1]$ which is universal in the sense of permutations with respect to the Walsh system for the classes $L^{0}[0,1]$ and $M[0,1]$? Question 3. Does Theorem 4 hold for the trigonometric system? Question 4. Does there exist a function $U\in L^{1}(0,1)$ which is universal in the sense of signs with respect to the Haar or Franklin system for some class $L^{p}[0,1]$, $p\in [0,1)$? Question 5. Does there exist a function $U\in L^{1}[0,2\pi)$ which is universal in the sense of permutations with respect to the trigonometric system for the classes $L^{p}[0,2\pi]$, $p\in(0,1)$? Question 6. Do there exist an orthonormal system $\{\varphi_k(x)\}_{k=0}^{\infty}$ of bounded functions and a function $U\in L^{1}[0,1)$ which is universal with respect to the system $\{\varphi_k(x)\}_{k=0}^{\infty}$ for some class $L^{p}[0,1]$, $p\in [0,1)$? The author is grateful to B. S. Kashin for his interest in this study and useful comments. § 2. Auxiliary resultsThe Walsh–Paley system $W=\{W_n(x)\}$ is defined by (see [29])
$$
\begin{equation}
W_0(x)=1\quad\text{and} \quad W_n(x)=\prod_{s=1}^{k}r_{m_s}(x), \qquad n=\sum_{s=1}^{k}2^{m_s}, \quad m_1>m_{2}>\dots >m_s,
\end{equation}
\tag{2.1}
$$
where $\{r_k(x)\}_{k=0}^{\infty}$ is the Rademacher system:
$$
\begin{equation*}
\begin{gathered} \, r_0(x)= \begin{cases} 1, &x\in\biggl[0,\dfrac{1}{2}\biggr), \\ r-1, &x\in\biggl[\dfrac{1}{2},1\biggr), \end{cases} \\ r_0(x+1)=r_0(x)\quad\text{and} \quad r_k(x)=r_0(2^{k}x), \quad k=1,2,\dots\,. \end{gathered}
\end{equation*}
\notag
$$
The Walsh–Paley system is a system of functions popular with authors and studied extensively. An important property of this system is that it forms an orthogonal basis of $L^{p}[0,1)$, $p\in(1,\infty)$ (see [30] and [31]). We need some definitions. Let $|E|$ be the Lebesgue measure of the measurable set $E\subseteq [0,1)$. We partition the half-open interval $[0,1)$ into $2^{m}$ equal subintervals $[(k-1)/2^{m},k/2^{m})$, $k\in [1,2^{m}]$, which we call dyadic intervals. Let
$$
\begin{equation}
\chi_{E}(x)= \begin{cases} 1, &x\in E, \\ 0, &x\notin E, \end{cases}
\end{equation}
\tag{2.2}
$$
be the characteristic function of the set $E$, and let be the Fourier–Walsh coefficients of the function $g\in L^{1}(0,1)$. We also set For an arbitrary positive number $\delta$ and a natural number $n$ we have
$$
\begin{equation}
|S_{n}(x,g)| <\frac{2}{\delta}\int_{c}^{d}|g(t)|\,dt \quad \forall\, x\notin[ c-\delta,d+\delta],
\end{equation}
\tag{2.5}
$$
where $g(t)$ is an arbitrary integrable function vanishing outside $(c,d)$. In the proofs of the main lemmas we use the following well-known properties of the Walsh system (see [31]):
$$
\begin{equation}
W_{i}(x)W_j(2^{s}x)=W_{j2^{s}+i}(x) \quad\text{for } 0\leqslant i <2^{s} \text{ (see (2.1))},
\end{equation}
\tag{2.6}
$$
$$
\begin{equation}
\biggl|\sum_{k=0}^{n}W_k(x)\biggr|\leqslant\frac{1}{x},
\end{equation}
\tag{2.7}
$$
$$
\begin{equation}
\sum_{k=0}^{2^{m}-1}W_k(x)= \begin{cases} 2^{m}, & x\in[0,2^{-m}), \\ 0, & x\in[2^{-m},1). \end{cases}
\end{equation}
\tag{2.8}
$$
From these inequalities, for all natural numbers $1\leqslant M<N\leqslant2^{n}$ we have
$$
\begin{equation}
\int_0^{1}\biggl|\sum_{k=M}^{N}W_k(x)\biggr|\, dx \leqslant\int_0^{2^{-n}}2^n\,dx+2\int_{2^{-n}}^{1}\frac{1}{x}\,dx\leqslant3n,
\end{equation}
\tag{2.9}
$$
$$
\begin{equation}
\sum_{k=2^{m}}^{2^{m+1}-1}W_k(x)= \begin{cases} 2^{m}, & x\in[ 0,2^{-m-1}), \\ -2^{m}, & x\in(2^{-m-1},2^{-m}), \\ 0, & x\in(2^{-m},1]. \end{cases}
\end{equation}
\tag{2.10}
$$
We also need the following result from [32]. Lemma 1. For each dyadic interval $\Delta:=[(k-1)/2^{\sigma},k/2^{\sigma})$, $k\in [1,2^{\sigma}]$, and any natural number $ m >\sigma$ such that $m-\sigma$ is even there exist measurable sets $E^{+}, E^{-}\subset \Delta$ and a polynomial in the Walsh system such that $E^{+}$ and $ E^{-}$ are finite unions of dyadic intervals, and § 3. Proofs of the main lemmasThe proofs of the main lemmas depend on some constructions from [33] and [34] (for the reader’s convenience we present the detailed proofs). Lemma 2. Let $n_0\in\mathbb{N}$, and let $\Delta =[(k-1)/2^l,k/2^l)\subset[2^{-n_0},1)$, $l\geqslant n_0$, be a dyadic interval. Then for all numbers $\eta\in(0,1)$ and $\gamma\neq0$ and all natural numbers $\lambda$ and $\nu$, $\lambda<\nu$, there exist measurable sets $G\subset E\subset\Delta$ and polynomials
$$
\begin{equation*}
U(x)=\sum_{k=2^{n_0}}^{{2^{n}}-1}b_kW_k(x)\quad\textit{and} \quad P(x)=\sum_{k=2^{n_0}}^{{2^{n}}-1}\delta_kb_kW_k(x), \quad\delta_k =\pm1,
\end{equation*}
\notag
$$
in the Walsh system such that
$$
\begin{equation*}
\begin{aligned} \, &(1)\quad |E|=(1-2^{-\nu})|\Delta|, \qquad |G|=|\Delta|(1-2^{-\lambda}), \\ &(2)\quad 0<b_{k+1}\leqslant b_k<\eta\ \textit{ for } k\in[2^{n_0},2^{n}-1), \\ &(3)\quad U(x)\cdot\chi_{[2^{-n_0},1]}(x)=0, \\ &(4)\quad P(x)= \begin{cases} \gamma, & x\in E, \\ 0, & x\in[2^{-n_0},1)\setminus\Delta, \end{cases} \\ &(5)\quad \int_0^{1}|U(x)|\,dx\leqslant\max_{2^{n_0}\leqslant M<2^{n}} \int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}b_kW_k(x)\biggr|\, dx<\eta, \\ &(6)\quad \max_{2^{n_0}\leqslant M<2^{n}}\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr|\, dx <A_1|\gamma|\,|\Delta|, \\ &(7)\quad \max_{2^{n_0}\leqslant M<2^{n}}\biggl|\sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| <\begin{cases} A_12^{\lambda}|\gamma |+\eta,& x\in G, \\ \eta,& x\in[2^{-n_0+1},1]\setminus\Delta, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
where $A_1$ is a constant. Proof. We partition $\Delta$ into a union of dyadic intervals $\Delta_{i}^{(1)}$, $i\in[1,N_1]$, such that
$$
\begin{equation}
|\Delta_{i}^{(1)}|=2^{-l_1}, \qquad i\in[1,N_1], \qquad N_1=2^{(l_1-l)},
\end{equation}
\tag{3.2}
$$
where the natural number $l_1$ satisfies ($A_0$ is the constant from Lemma 1).
We define recursively sets $E_1^{(-)}\supset E_{2}^{(-)}\supset\dots\supset E_s^{(-)}\supset\dotsb$, integers $l_1<l_{2}<\dots<l_s<\dotsb$ and $m_1<m_{2}<\dots<m_s<\dotsb$, and polynomials $\{Q_j^{(1)}(x)\}_{j=1}^{\infty}$, $\{Q_j^{(\Diamond)}(x)\}_{j=1}^{\infty}$, $\{{Q}_j^{(2)}(x)\}_{j=1}^{\infty}$, $\{P_s^{(\Diamond)}(x)\}_{s=1}^{\infty}$ and $\{P_s(x)\}_{s=1}^{\infty}$ that satisfy certain conditions (see (3.19)–(3.43)). Assume that we have already constructed polynomials $P_1(x),\dots, P_{s-1}(x)$, sets $E_{s-1}^{(-)}\subset E_{s-2}^{(-)}\subset\dots\subset E_1^{(-)}\subset E_0^{(-)}=\Delta$, and numbers $l_1<l_{2}<\dots<l_{s-1}$, $ m_1<m_{2}<\dots<m_{s-1}$ that for all $1\leqslant j\leqslant s-1$ satisfy and
$$
\begin{equation}
P_j(x)={Q}_j^{(1)}(x)+{Q}_j^{(2)}(x)+P_j^{(\Diamond)}(x)=\sum_{k=2^{n_{j-1}+1}}^{2^{n_j+1}-1} a_kW_k(x) \quad \forall\, x\in E_j^{(-)},
\end{equation}
\tag{3.6}
$$
where $E_j^{(-)}$ is a finite union of dyadic intervals of measure $ 2^{-j}|\Delta|=2^{-j-l}$, and
$$
\begin{equation}
|a_k|\geqslant|a_{k+1}|\geqslant\dots\geqslant|a_{2^{n_j+1}-1}|=2^{s-1} |\gamma|2^{-(\sigma_j+m_j)/2} \quad\forall\, k\in[ 2^{n_{j-1}+1},2^{n_j+1})
\end{equation}
\tag{3.7}
$$
(for $j=1$, $2^{n_{j-1}+1}$ is set to be $2^{n_0}$).
Let $l_s$ and $N_s$ be natural numbers such that and and let
$$
\begin{equation}
E_{s-1}^{(-)} =\bigcup_{i=1}^{N_s}\Delta_{i}^{(s)},
\end{equation}
\tag{3.10}
$$
where $\Delta_j^{(s)}$ is a dyadic interval of measure
$$
\begin{equation}
|\Delta_{i}^{(s)}|=2^{-l_s} \quad\forall\, i\in[1,N_s].
\end{equation}
\tag{3.11}
$$
From (3.3), (3.5) and (3.8) we obtain It is clear that (see also (3.3))
$$
\begin{equation}
|\gamma|2^{-l_s}\leqslant|\gamma|2^{-2(s-1)-l_1}\leqslant\min\biggl\{\frac{\eta }{2^{2(s-1)}};\frac{|\gamma|2^{-l}}{2^{2s}}\biggr\}.
\end{equation}
\tag{3.13}
$$
Let $m_s$ be a natural number such that
$$
\begin{equation}
m_s\geqslant\max\{N_s+2l+2s;n_{s-1}+1\}, \qquad 2^{m_s/2}\geqslant(m_s+l_s\}2^{l_s/2},
\end{equation}
\tag{3.14}
$$
and $m_s-l_s$ is even.
We set and
$$
\begin{equation}
Q_s^{(1)}(x)=\sum_{k=2^{n_{s-1}+1}}^{2^{n_s}-1}a_kW_k(x) =\frac{2^{s-1}|\gamma|2^{-l_s}}{n_s}\sum_{k=2^{n_{s-1}+1}}^{2^{n_s}-1}W_k(x).
\end{equation}
\tag{3.16}
$$
From (3.3), for $s=1$ we obtain
$$
\begin{equation}
|a_k|=\frac{|\gamma|2^{-l_1}}{n_1}\leqslant|\gamma|2^{-l_1}\leqslant \eta \quad \forall\, k\in[2^{n_0},2^{n_1}),
\end{equation}
\tag{3.17}
$$
and for $s>1$, from (2.7), (3.8) and (3.15) we see for $ k\in[2^{n_{s-1}+1},2^{n_s})$ that
$$
\begin{equation}
|a_k|=\frac{2^{s-1}|\gamma|2^{-l_s}}{n_s}\leqslant2^{s-2}|\gamma |2^{-(l_{s-1}+m_{s-1})/2}=|a_{2^{n_{s-1}+1}-1}| \quad\forall\, k\in[2^{n_{s-1}+1},2^{n_s}),
\end{equation}
\tag{3.18}
$$
that is, the coefficients of the polynomial ${Q}_s^{(1)}(x)$ have smaller absolute values than those of $P_{s-1}(x)$.
From (2.7)–(2.9), (3.12), (3.15), (3.16) and (3.18), for the polynomial ${Q}_s^{(1)}(x)$ we obtain
$$
\begin{equation}
{Q}_s^{(1)}(x)=0 \qquad\forall\, x\geqslant2^{-n_0},
\end{equation}
\tag{3.19}
$$
$$
\begin{equation}
\int_0^{1}|Q_s^{(1)}(x)|\,dx \leqslant2\frac{2^{s-1}|\gamma|2^{-l_s}}{n_s} \leqslant\min\biggl\{\frac{\eta}{2^{(s-1)}};\frac{|\gamma|2^{-l}}{2^{s}}\biggr\} \leqslant2^{-s}|\gamma|\,|\Delta|
\end{equation}
\tag{3.20}
$$
and
$$
\begin{equation}
\begin{split} \max_{2^{n_{s-1}+1}\leqslant M<2^{n_s}}\int_0^{1} \biggl|\sum_{k=2^{n_{s-1}+1}}^{M}a_kW_k(x)\biggr|\, dx &\leqslant 3n_s \frac{2^{s-1}|\gamma|2^{-l_s}}{n_s} \\ &\leqslant\min\biggl\{\frac{\eta}{2^{(s+1)}};\frac{|\gamma|\,|\Delta|}{2^{s-1}}\biggr\}, \end{split}
\end{equation}
\tag{3.21}
$$
$$
\begin{equation}
\max_{2^{n_{s-1}+1}\leqslant M<2^{n_s}} \biggl| \sum_{k=2^{n_{s-1}+1}}^{M}a_kW_k(x)\biggr| \leqslant\frac{2^{s-1}|\gamma|2^{-l_s}}{n_s}\frac{1}{x}\leqslant\frac{\eta}{2^{(s+1)}} \quad\forall\, x>[2^{-n_0},1).
\end{equation}
\tag{3.22}
$$
For each $j\in[1,N_s]$ an appeal to Lemma 1 for $m=m_s$ and $\Delta=\Delta_j^{(s)}$ gives the polynomial
$$
\begin{equation*}
\sum_{k=2^{m_s}}^{2^{m_s+1}-1}\beta_k^{(j)}W_k(x)= \begin{cases} \pm1, & x\in e_j^{\pm}\subset\Delta_j^{(s)}, \\ 0, & x\notin\Delta_j^{(s)}, \end{cases} \qquad |e_j^{+}|=|e_j^{-}|=\frac{|\Delta_j^{(s)}|}{2},
\end{equation*}
\notag
$$
where $e_j^{+}$ and $ e_j^{-}$ are finite unions of dyadic intervals, and we have
$$
\begin{equation*}
\begin{gathered} \, |\beta_k^{(j)}|=2^{-(m_s +l_s)/2}, \qquad k=2^{m},\dots, 2^{m+1}-1, \\ \max_{0\leqslant M<2^{n}}\biggl| \sum_{k=2^{n_0}}^{M}\beta_k^{(j)}W_k(x)\biggr| <2^{-(m_s -l_s)/2}, \qquad x\notin\Delta _j^{(s)}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\max_{0\leqslant M<2^{n}}\biggl| \sum_{k=2^{n_0}}^{M}\beta_k^{(j)}W_k(x)\biggr| <A_0, x\in\Delta_j^{(s)}
\end{equation*}
\notag
$$
($A_0$ is the constant from Lemma 1). Setting
$$
\begin{equation}
P_s^{(j)}(x):=\sum_{k=2^{m_s}}^{2^{m_s+1}-1}2^{s-1}\gamma\beta _k^{(j)}W_k(x)=\sum_{k=2^{m_s}}^{2^{m_s+1}-1}a_kW_k(x),
\end{equation}
\tag{3.23}
$$
we have
$$
\begin{equation}
\nonumber \sum_{k=2^{m_s}}^{2^{m_s+1}-1}2^{s-1}\gamma\beta_k^{(j)} W_k(x)= \begin{cases} \pm2^{s-1}\gamma, & x\in e_j^{\pm}\subset\Delta_j^{(s)}, \\ 0, & x\notin\Delta_j^{(s)}, \end{cases}
\end{equation}
\notag
$$
$$
\begin{equation}
\max_{2^{m_s}\leqslant M<2^{m_s+1}} \biggl| \sum_{k=2^{m_s}}^{M}2^{s-1}\gamma\beta_k^{(j)}W_k(x)\biggr| \leqslant A_02^{s-1}|\gamma| \quad\forall\, x\in\Delta_j^{(s)} ,
\end{equation}
\tag{3.24}
$$
$$
\begin{equation}
\nonumber P_s^{(j)}(x)=0, \qquad \max_{2^{m_s}\leqslant M<2^{m_s+1}}\biggl| \sum_{k=2^{m_s}}^{M}2^{s-1}\gamma\beta_k^{(j)}W_k(x)\biggr|
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad\qquad\qquad\leqslant 2^{s-1}|\gamma|2^{-(m_s -l_s)/2} \quad\forall\, x\notin\Delta_j^{(s)},
\end{equation}
\tag{3.25}
$$
and
$$
\begin{equation}
2^{s-1} |\gamma\beta_k^{(j)}|=2^{s-1}|\gamma|2^{-(l_s+m_s)/2} \leqslant\frac{2^{s-1}|\gamma|2^{-l_s}}{l_s+m_s}\leqslant\frac{2^{s-1}|\gamma|2^{-l_s}}{n_s}.
\end{equation}
\tag{3.26}
$$
We also set
$$
\begin{equation}
P_s^{(\Diamond)}(x):=\sum_{j=1}^{N_s}P_s^{(j)}(x)W_{2^{n_s}+(j-1)2^{m_s+1}}(x)
\end{equation}
\tag{3.27}
$$
and
$$
\begin{equation}
E_s^{(-)}:=\bigl\{x\in E_{s-1}^{(-)} ;\gamma P_s^{(\Diamond )}(x)<0\bigr\} \quad\text{and} \quad E_s^{(+)}:=\bigl\{x\in E_{s-1}^{(-)} ;\gamma P_s^{(\Diamond)}(x)>0\bigr\}.
\end{equation}
\tag{3.28}
$$
$$
\begin{equation}
P_s^{(\Diamond)}(x)=0 \quad\forall\, x\notin E_{s-1}^{(-)},
\end{equation}
\tag{3.29}
$$
$$
\begin{equation}
P_s^{(\Diamond)}(x)=2^{s-1}\gamma \quad \forall\, x\in E_s^{(+)},
\end{equation}
\tag{3.30}
$$
$$
\begin{equation}
P_s^{(\Diamond)}(x)=-2^{s-1}\gamma \quad\forall\, x\in E_s^{(-)};
\end{equation}
\tag{3.31}
$$
$E_s^{(-)}$ and $ E_s^{(+)} $ are finite unions of dyadic intervals, and
$$
\begin{equation}
|E_s^{(+)}|= |E_s^{(-)}|=\frac{1}{2} |E_{s-1}^{(-)}|=\frac{1}{2^{s}}|\Delta|.
\end{equation}
\tag{3.32}
$$
It is clear that (see (3.10), (3.22), (3.23), (3.25), (3.27) and (3.32))
$$
\begin{equation}
\begin{aligned} \, \notag \int_0^{1}|P_s^{(\Diamond)}(x)|\,dx &=\sum_{j=1}^{N_s}\int_0^{1}|P_s^{(j)}(x)|\,dx =\sum_{j=1}^{N_s}2^{s-1}|\gamma|\,|\Delta_j^{(s)} | \\ &=2^{s-1}|\gamma|\,|E_{s-1}^{(-)}|=|\gamma|\,|\Delta|. \end{aligned}
\end{equation}
\tag{3.33}
$$
We set
$$
\begin{equation}
{Q}_s^{(\diamondsuit)}(x):=\sum_{k=0}^{2^{m_s}-1}2^{s-1}\gamma2^{-(l_s+m_s)/2}W_k(x)
\end{equation}
\tag{3.34}
$$
and
$$
\begin{equation}
{Q}_s^{(2)}(x):=\sum_{j=1}^{N_s}{Q}_s^{(\diamondsuit)}(x)W_{2^{n_s}+(j-1)2^{m_s+1}}(x).
\end{equation}
\tag{3.35}
$$
By the definition of the polynomial ${Q}_s^{(2)}(x)$, from (2.8), (3.13), (3.14), (3.34), and (3.35) we obtain and
$$
\begin{equation}
\int_0^{1}|{Q}_s^{(2)}(x)|\,dx \leqslant N_s2^{s-1}|\gamma|2^{-(l_s+m_s)/2} =|\gamma|2^{-(m_s-l_s)/2-l}\leqslant2^{-s}|\gamma|\,|\Delta|.
\end{equation}
\tag{3.37}
$$
Consider the polynomials
$$
\begin{equation}
P_s(x):={Q}_s^{(1)}(x)+{Q}_s^{(2)}(x)+P_s^{(\Diamond)}(x) =\sum_{k=2^{n_{s-1}+1}}^{2^{n_s+1}-1}a_kW_k(x)
\end{equation}
\tag{3.38}
$$
and
$$
\begin{equation}
U_s(x):=\sum_{k=2^{n_{s-1}+1}}^{2^{n_s+1}-1}|a_k|W_k(x)={Q} _s^{(1)}(x)+\sum_{k=2^{n_s}}^{2^{n_s+1}-1}|a_k|W_k(x).
\end{equation}
\tag{3.39}
$$
It is clear that the absolute value of $a_k$ is $2^{s-1}|\gamma|2^{-(m_s +l_s)/2}$ for $k\in[2^{n_s},2^{n_s+1})$ (see (2.6)). Hence for the coefficients of the polynomial $P_s(x)$ we have (see also (3.18) and (3.26))
$$
\begin{equation}
|a_{2^{n_{s-1}+1}-1}|\geqslant|a_{2^{n_{s-1}+1}}|=\dots =|a_{2^{n_s}-1}|\geqslant|a_{2^{n_s}}|=\dots =|a_{2^{n_s+1}-1}|>0.
\end{equation}
\tag{3.40}
$$
From (2.8), (3.25)–(3.27), (3.34), (3.35) and (3.39) we obtain An appeal to (2.9), (3.13)–(3.17), (3.21) and (3.26) shows that
$$
\begin{equation}
\begin{aligned} \, \notag &\max_{2^{n_s}\leqslant M<2^{n_s+1}}\int_0^{1} \biggl|\sum_{k=2^{n_{s-1}+1}}^{M}|a_k|W_k(x)\biggr|\, dx \leqslant\frac{\eta}{2^{s+2}}+3n_s2^{s-1}|\gamma|2^{-(m_s +l_s)/2} \\ &\qquad\leqslant\frac{\eta}{2^{s+2}}+3n_s\frac{2^{s-1}|\gamma|2^{-l_s}}{m_s +l_s}. \end{aligned}
\end{equation}
\tag{3.42}
$$
Hence by (3.15), (3.18), (3.21) and (3.42) we have
$$
\begin{equation}
\max_{M<2^{n_s+1}}\int_0^{1}\biggl| \sum_{k=2^{n_{s-1}+1}}^{M}|a_k|W_k(x)\biggr|\, dx \leqslant\frac{\eta}{2^{s+1}}, \qquad |a_k|>0.
\end{equation}
\tag{3.43}
$$
So we have defined recursively the sets $E_1^{(-)}\supset E_{\lambda}^{(-)}\supset\dots \supset E_j^{(-)}\supset\dots \supset\dots\supset E_{\nu}^{(-)}\dotsb$ and the polynomials $\{{Q}_j^{(1)}(x)\}_{j=1}^{\infty}$, $\{{Q}_j^{(\Diamond)}(x)\}_{j=1}^{\infty}$, $\{{Q}_j^{(2)}(x)\}_{j=1}^{\infty}$, $\{P_s^{(\Diamond)}(x)\}_{s=1}^{\infty}$, $\{P_s(x)\}_{s=1}^{\infty}$ satisfying conditions (3.19)–(3.43) (the natural numbers $\lambda$ and $\nu$, $\lambda<\nu$, have already been defined: see Lemma 1).
Consider the sets
$$
\begin{equation}
E:=\Delta\setminus E_{\nu}^{(-)}, \qquad G:=\Delta\setminus E_{\lambda}^{(-)}
\end{equation}
\tag{3.44}
$$
and the polynomials
$$
\begin{equation}
U(x) :=\sum_{s=1}^{\nu}U_s(x)=\sum_{k=2^{n_0}}^{2^{n_{\nu}+1}-1}b_kW_k(x) =\sum_{k=2^{n_0}}^{2^n-1}|a_k|W_k(x)
\end{equation}
\tag{3.45}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag P(x) &:=\sum_{j=1}^{\nu}{Q}_j^{(1)}(x)+\sum_{j=1}^{\nu}{Q}_j^{(2)}(x) +\sum_{j=1}^{\nu}P_j^{(\Diamond)}(x) \\ & =\sum_{k=2^{n_0}}^{2^{n_s+1}-1}a_kW_k(x)=\sum_{k=2^{n_0}}^{2^n-1} \delta_kb_kW_k(x), \end{aligned}
\end{equation}
\tag{3.46}
$$
where
$$
\begin{equation}
\delta_k:=\operatorname{sign} \{a_k\}=\pm1\quad\text{and} \quad b_k:=|a_k|, \qquad k\in[2^{n_0},2^n), \qquad n:=n_{\nu}+1.
\end{equation}
\tag{3.47}
$$
It is clear that (see (3.2), (3.32) and (3.41)–(3.45))
$$
\begin{equation*}
\begin{gathered} \, |E|=(1-2^{-\nu})|\Delta|, \qquad|G|=(1-2^{-\lambda})|\Delta|, \\ U(x)=0 \quad\forall\, x\in[2^{-n_0} ,1)\quad\text{and} \quad \max_{M<2^n}\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}b_kW_k(x)\biggr|\, dx\leqslant\eta. \end{gathered}
\end{equation*}
\notag
$$
From (3.23) and (3.27)–(3.31), for each $s\leqslant\nu$ we obtain
$$
\begin{equation}
\sum_{j=1}^{s}P_j^{(\Diamond)}(x)= \begin{cases} \gamma, & x\in\Delta\setminus E_s^{(-)}, \\ -(2^{s-1}-1)\gamma, & x\in E_s^{(-)}, \\ 0, & x\notin\Delta. \end{cases}
\end{equation}
\tag{3.48}
$$
Using (3.17)–(3.19), (3.29), (3.32) and (3.35) we show that the polynomials $U(x)$ and $P(x)$ and the sets $E$ and $G$ satisfy conditions (1)–(5), (7) and (8) in Lemma 2. It is also clear that for each $s\leqslant\nu$
$$
\begin{equation}
\max_{s\leqslant\nu}\int_0^{1}\biggl| \sum_{j=1}^{s}P_j^{(\Diamond)}(x)\biggr|\, dx \leqslant|\gamma|\,|\Delta\setminus E_{\nu}^{(-)}|+2^{\nu -1}|\gamma|\,|E_{\nu}^{(-)}|\leqslant2|\gamma|\,|\Delta|.
\end{equation}
\tag{3.49}
$$
Let $M\in[2^{n_0},2^n)$ be a natural number. For some $s$, $1\leqslant s\leqslant\nu$, we have $M\in[ 2^{1+n_{s-1}},2^{n_s})$ (for $s=1$ $2^{1+n_{s-1}}$ is replaced by $2^{n_0}$). For $M\in[2^{n_{s-1}+1},2^{n_s})$, $M<2^{n_\nu}$, $1\leqslant s\leqslant\nu$, the polynomial (see (3.46) and (3.47)) has the form
$$
\begin{equation}
\sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)= \sum_{j=1}^{s-1}{Q}_j^{(1)}(x) +\sum_{j=1}^{s-1}{Q}_j^{(2)}(x)+\sum_{k=1}^{s-1}P_j^{(\Diamond)}(x) +\sum_{k=2^{n_{s-1}+1}}^{M}\delta_kb_kW_k(x),
\end{equation}
\tag{3.51}
$$
and now from (3.20) we see that the $L^{1}[0,1)$-norm of the first term is less than $|\gamma|\,|\Delta|$. A similar estimate also holds for the norm of the second term (see (3.37)). Hence by (3.21) and (3.49)
$$
\begin{equation*}
\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr|\, dx\leqslant5|\gamma|\,|\Delta|.
\end{equation*}
\notag
$$
If $M>2^{n_s}$, then $M\in[2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s})$ for some integer $j\in[1,2^{n_s-m_s}]$. Therefore, the polynomial (3.50) reads
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x) = \sum_{j=1}^{s}{Q}_j^{(1)}(x)+\sum_{j=1}^{s-1}{Q}_j^{(2)}(x) +\sum_{k=1}^{s-1}P_j^{(\Diamond)}(x) \\ &\qquad +\sum_{j=2^{n_s}}^{2^{n_s}+j2^{m_s}-1}\delta_kb_kW_k(x) +\sum_{k=2^{n_s}+j2^{m_s}-1}^{M}\delta_kb_kW_k(x). \end{aligned}
\end{equation}
\tag{3.52}
$$
Hence, for $M\in[2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s})$,
$$
\begin{equation}
\begin{aligned} \, \notag \int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr|\, dx &\leqslant\int_0^{1}\biggl| \sum_{j=1}^{s}{Q}_j^{(1)}(x)\biggr|\, dx +\int_0^{1}\biggl| \sum_{j=1}^{s-1}{Q}_j^{(2)}(x)\biggr|\, dx \\ &\qquad+\int_0^{1}\biggl| \sum_{j=1}^{s-1}P_j^{(\Diamond)}(x)\biggr|\, dx + \int_0^{1}\biggl| \sum_{k=2^{n_s}}^{2^{n_s}+j2^{m_s}-1}a_kW_k(x)\biggr|\, dx \notag \\ &\qquad+\int_0^{1}\biggl|\sum_{k=2^{n_s}+j2^{m_s}}^{M}a_kW_k(x)\biggr|\, dx. \end{aligned}
\end{equation}
\tag{3.53}
$$
As before, the sum of the first three terms is less than $4|\gamma|\,|\Delta|$. Proceeding as in (3.33) and (3.37) we see that the fourth term is less than $2|\gamma|\,|\Delta|$. For the fourth term we have (see (2.10), (3.34) and (3.35))
$$
\begin{equation}
\sum_{j=2^{n_s}}^{2^{n_s}+j2^{m_s}-1}\delta_k b_kW_k(x)=0, \qquad x\in[2^{-n_0},1).
\end{equation}
\tag{3.54}
$$
As concerns the last term, there are two cases to consider. If $j$ is even, then from the definition of the polynomial ${Q}_j^{(2)}(x)$ and relations (2.6), (2.7), (2.9), (3.13) and (3.14) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \int_0^{1}\biggl| \sum_{k=2^{n_s}+j2^{m_s}}^{M}\delta_kb_kW_k(x)\biggr|\, dx &=2^{s-1}|\gamma|2^{-(m_s+l_s)/2}\int_0^{1} \biggl| \sum_{k=0}^{M-2^{n_s}-j2^{m_s}}W_k(x)\biggr|\, dx \\ & \leqslant 3m_s2^{s-1}|\gamma|2^{-(m_s +l_s)/2}\leqslant5|\gamma|\,|\Delta|, \end{aligned}
\end{equation}
\tag{3.55}
$$
and for $x\in[2^{-n_0},1)$
$$
\begin{equation}
\begin{aligned} \, \notag \biggl| \sum_{k =2^{n_s}+j2^{m_s}}^{M}\delta_kb_kW_k(x)\biggr| &=2^{s-1}|\gamma|2^{-(m_s +l_s)/2} \biggl| \sum_{k=0}^{M-2^{n_s}-j2^{m_s}}W_k(x)\biggr| \\ & \leqslant2^{r-1}|\gamma|2^{-(m_s +l_s)/2}\frac{1}{x}\leqslant \eta. \end{aligned}
\end{equation}
\tag{3.56}
$$
If $j$ is odd, then in view of (3.6), (3.23)–(3.25) and the definition of the polynomial $P_s^{(j)}(x)$
$$
\begin{equation}
\begin{aligned} \, \notag & \int_0^{1}\biggl| \sum_{k=2^{n_s}+j2^{m_s}}^{M}\delta_kb_kW_k(x)\biggr|\, dx =\int_{E_{s-1}^{(-)}}^{1}+\int_{[0,1)\setminus E_{s-1}^{(-)}}^{1} \\ \notag &\qquad\leqslant A_0|2^{s-1}|\gamma|\,|E_{s-1}^{(-)}|+2^{s-1}|\gamma|2^{-(m_s -l_s)/2} \\ &\qquad \leqslant A_0|\gamma|\,|\Delta|+2^{s-1}|\gamma|2^{-(s+l)}\leqslant(A_0+1)|\gamma|\,|\Delta|. \end{aligned}
\end{equation}
\tag{3.57}
$$
Hence for $x\in[2^{-n_0},1)$
$$
\begin{equation}
\begin{aligned} \, \notag \biggl| \sum_{k=2^{n_s}+j2^{m_s}}^{M}\delta_kb_kW_k(x)\biggr| &\leqslant A_02^{s-1}|\gamma|\chi_{E_{s-1}^{(-)}}(x)+2^{s-1}|\gamma|2^{-(m_s -l_s)/2}\chi_{[ 0,1)\setminus E_{s-1}^{(-)}}(x) \\ & \leqslant(A_0+1)2^{s-1}|\gamma|, \end{aligned}
\end{equation}
\tag{3.58}
$$
where $A_0$ is the constant from Lemma 1. Therefore, in view of (3.53), (3.55) and (3.57), for any natural number $M\in[2^{n_0},2^n)$ we have
$$
\begin{equation*}
\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr|\, dx \leqslant A|\gamma|\,|\Delta|, \qquad A=A_0+1.
\end{equation*}
\notag
$$
Now we verify condition (7) in Lemma 2. Let $M <2^{n_{\lambda}}$. Using (3.19), (3.22), (3.36), (3.44), (3.48), (3.51) and (3.58) for $M\in [2^{n_{s-1}+1},2^{n_s})$, $s\leqslant\lambda$, for all $x\in [2^{-n_0},1)$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| &\leqslant2^{s-1}|\gamma|\chi_{E_{s-1}^{(-)}}(x)+|\gamma|\chi_{[0,1)\setminus E_{s-1}^{(-)}}(x)+A_02^{r-1}|\gamma|\chi_{E_{s-1}^{(-)}}(x) \\ &\qquad+2^{s-1}|\gamma|2^{-(m_s -l_s)/2}\chi_{[0,1)\setminus E_{s-1}^{(-)}}(x)+\frac{\eta}{2^{(\lambda+1)}} \notag \\ &\leqslant A_02^{\lambda} |\gamma|, \end{aligned}
\end{equation}
\tag{3.59}
$$
and if $M\in[2^{n_s},2^{n_{\lambda}})$, $s\leqslant\lambda$, then $M\in[2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s})$ for some $j\in [1,2^{n_s-m_s}]$.
Therefore, in view of (3.54), (3.56), (3.58) and (3.59), for any natural number $M\in[2^{n_0},2^n)$ and all $x\in [2^{-n_0},1)$,
$$
\begin{equation}
\begin{aligned} \, \biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| &\leqslant2^{s-1}|\gamma|+A_02^{s-1}|\gamma|+|\gamma|+2^{s-1}|\gamma |2^{-(m_s -l_s)/2}+\eta \notag \\ &\leqslant(A_0+2)2^{\lambda}|\gamma|. \end{aligned}
\end{equation}
\tag{3.60}
$$
If $M \geqslant2^{n_{\lambda}}$, then $M\in [2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s})$ for some integers $s\geqslant\lambda+1$ and $j\in[1,2^{n_s-m_s}]$. Next, we have (see (3.19), (3.29), (3.36) and (3.44))
$$
\begin{equation*}
{Q}_s^{(1)}(x) = {Q}_s^{(2)}(x)=0 \quad \forall\, x>2^{-n_0}, \quad \forall\, s=1,2,\dots,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sum_{j=\lambda+1}^{r-1}P_j^{(\Diamond)}(x)=0 \quad \forall\, x\in G,
\end{equation*}
\notag
$$
and so by (3.52)
$$
\begin{equation*}
\begin{aligned} \, & \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)= 0+\sum_{j=1}^{\lambda}P_j^{(\Diamond)}(x)+\sum_{j=2^{n_s}}^{2^{n_s}+j2^{m_s}-1}\delta_kb_kW_k(x) \\ &\qquad +\sum_{k=2^{n_s}+j2^{m_s}-1}^{M}\delta_kb_kW_k(x), \qquad M\in[2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s}). \end{aligned}
\end{equation*}
\notag
$$
Now from (3.48), (3.54), (3.56), (3.56) and (3.58), for all $x\in G$ we have
$$
\begin{equation*}
\begin{aligned} \, \biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| &<2^{\lambda}|\gamma|+2^{s-1}|\gamma|2^{-(m_s+l_s)/2}\frac{1}{x}+2^{s-1}|\gamma|2^{-(m_s -l_s)/2} \\ &\leqslant(A_0+2)2^{\lambda}|\gamma|, \end{aligned}
\end{equation*}
\notag
$$
and for $x\in[2^{-n_0},1)\setminus \Delta$,
$$
\begin{equation*}
\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| <0+2^{s-1}|\gamma|2^{-(m_s+l_s)/2}\frac{1}{x}+2^{s-1}|\gamma|2^{-(m_s -l_s)/2}\leqslant\eta.
\end{equation*}
\notag
$$
Using (3.59), (3.60) and the last two relations we find that
$$
\begin{equation*}
\biggl|\sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr| < \begin{cases} A_12^{\lambda}|\gamma|, &x\in G, \\ \eta, &x\in [2^{-n_0},1]\setminus\Delta, \end{cases}
\end{equation*}
\notag
$$
where $A_1=A_0+2$. This proves Lemma 2. Lemma 3. Let $n_0\in\mathbb{N}$ and $\varepsilon \leqslant\delta\in(0,1)$, let $f(x)=\sum_{m=1}^{\widetilde{\nu}_0}\widetilde{\gamma}_{m}\chi_{\widetilde{\Delta}_{m}}(x)$ be a step function such that $\widetilde{\gamma}_{m}\neq0$, and let $\{\widetilde{\Delta}_{m}\}_{m=1}^{\widetilde{\nu}_0}$ be disjoint binary dyadic intervals such that $\sum_{m=1}^{\widetilde{\nu}_0}|\widetilde{\Delta}_{m}|=1$. Then there exist measurable sets $ G\subset E\subset[2^{-n_0},1)$ and polynomials Proof. Let the numbers $\nu$ and $\lambda$ be defined by
Consider the function We split $[0,1]$ into disjoint dyadic intervals of the same length $\{\Delta_j\}$ so that $|\Delta_j|\leqslant\min\{|\widetilde{\Delta}_{m}|\}$ and $2^{-n_0}$ is not an interior point of these intervals. We write the function $f_0(x)$ as where $\gamma_j=\widetilde{\gamma}_{m}$ if $\Delta_j\subset\widetilde{\Delta}_{m}$.Applying Lemma 2 to each interval $\Delta_j$, $j\in[1,\mu]$, in succession and employing (3.61), for all $ j\in[1,\mu]$ we find sets $G_j\subset E_j\subset\Delta_j\subset[2^{-n_0},1]$ such that
$$
\begin{equation}
|E_j|=(1-2^{-\nu})|\Delta_j|>(1-\varepsilon)|\Delta_j|
\end{equation}
\tag{3.64}
$$
and
$$
\begin{equation}
|G_j|=(1-2^{-\lambda})|\Delta_j|>(1-\delta)|\Delta_j|
\end{equation}
\tag{3.65}
$$
and find polynomials
$$
\begin{equation}
U_j(x)=\sum_{k=2^{n_{j-1}}}^{{2^{n_j}}-1}a_k^{(j)}W_k(x)\quad\text{and} \quad P_j(x) =\sum_{k=2^{n_{j-1}}}^{{2^{n_j}}-1}\delta_k^{(j)}a_k^{(j)}W_k(x), \quad \delta_k^{(j)}=\pm1,
\end{equation}
\tag{3.66}
$$
in the Walsh system that satisfy
$$
\begin{equation}
\begin{cases} 0<a_{k+1}^{(1)}\leqslant a_k^{(1)}<\varepsilon, & k\in [2^{n_0},2^{n_1}-1), \\ 0<a_{k+1}^{(j)}\leqslant a_k^{(j)}<a_{2^{n_{j-1}}-1}^{(j-1)}, & k\in[2^{n_{j-1}},2^{n_j}-1),\quad j\in[2,\mu], \end{cases}
\end{equation}
\tag{3.67}
$$
$$
\begin{equation}
P_j(x)=\begin{cases} \gamma_j, & x\in E_j, \\ 0, & x\in[2^{-n_0},1]\setminus\Delta_j, \end{cases} \quad\text{if}\quad\Delta_j\subset[2^{-n_0},1],
\end{equation}
\tag{3.69}
$$
$$
\begin{equation}
\int_0^{1}|U_j(x)|\,dx\leqslant \max_{2^{n_{j-1}}\leqslant M<2^{n_j}} \int_0^{1}\biggl| \sum_{k=2^{n_{j-1}}}^{M}a_k^{(j)}W_k(x)\biggr|\, dx <\frac{\varepsilon}{2\mu}
\end{equation}
\tag{3.70}
$$
and
$$
\begin{equation}
\int_0^{1}|P_j(x)|\,dx\leqslant\max_{2^{n_{j-1}}\leqslant M<2^{n_j}} \int_0^{1}\biggl| \sum_{k=2^{n_{j-1}}}^{M}\delta_k^{(j)}a_k^{(j)}W_k(x)\biggr|\, dx <A_1|\gamma_j|\,|\Delta_j|,
\end{equation}
\tag{3.71}
$$
$$
\begin{equation}
\max_{2^{n_{j-1}}\leqslant M<2^{n_j}}\biggl| \sum_{k=2^{n_{j-1}}}^{M}\delta_k^{(j)}a_k^{(j)}W_k(x)\biggr| <\begin{cases} A_1 2^{\lambda}|\gamma_j|+\dfrac{\varepsilon}{2\mu},& x\in G_j, \\ \dfrac{\varepsilon}{2\mu},& x\in [2^{-n_0},1]\setminus\Delta_j, \end{cases}
\end{equation}
\tag{3.72}
$$
where $A_1$ is the constant from Lemma 2.
Consider the sets
$$
\begin{equation}
E:=\bigcup_{j=1}^{\mu}E_j\quad\text{and} \quad G=\bigcup_{j=1}^{\mu}G_j, \qquad G_j\subset E_j\subset\Delta_j\subset[2^{-n_0},1),
\end{equation}
\tag{3.73}
$$
and the polynomials
$$
\begin{equation}
U(x):=\sum_{j=1}^{\mu}U_j(x)=\sum_{k=2^{n_0}}^{2^{n_{\mu}-1}}a_kW_k(x)
\end{equation}
\tag{3.74}
$$
and
$$
\begin{equation}
P(x):=\sum_{j=1}^{\mu}P_j(x)=\sum_{k=2^{n_0}}^{2^{n_{\mu}-1}}\delta_ka_kW_k(x),
\end{equation}
\tag{3.75}
$$
where
$$
\begin{equation}
a_k:=a_k^{(j)}\quad\text{and} \quad\delta_k=\delta_k^{(j)}, \quad k\in[ 2^{n_{j-1}},2^{n_j}).
\end{equation}
\tag{3.76}
$$
From (3.61)–(3.69) and (3.73)–(3.76) we obtain
$$
\begin{equation*}
\begin{gathered} \, G\subset E\subset[2^{-n_0},1], \quad|E|>1-\varepsilon-2^{-n_0}, \quad G>1-\delta-2^{-n_0}, \\ 0<a_{k+1}\leqslant a_k<\varepsilon \quad\text{for}\quad k\in[2^{n_0},2^{n_{\mu}}-1), \\ U(x)\chi_{[2^{-n_0},1]}(x)=0 \end{gathered}
\end{equation*}
\notag
$$
and
Next, let $M$ be a natural number from $[2^{n_0},2^{n_{\mu}})$. Then $M\in[2^{n_{m-1}},2^{n_{m}})$ for some $m\in [1,\mu]$. Using (3.63) and (3.75)–(3.77), this gives
$$
\begin{equation}
\sum_{k=2^{n_0}}^{M}a_kW_k(x)=\sum_{j=1}^{m-1}U_j(x) +\sum_{k=2^{n_{m-1}}}^{M}a_k^{(m)}W_k(x)
\end{equation}
\tag{3.77}
$$
and
$$
\begin{equation}
\sum_{k=2^{n_0}}^{M}\delta_ka_kW_k(x)=\sum_{j=1}^{m-1}P_j(x) +\sum_{k=2^{n_{m-1}}}^{M}\delta_k^{(m)}a_k^{(m)}W_k(x).
\end{equation}
\tag{3.78}
$$
An appeal to (3.70), (3.71) and (3.74)–(3.78) shows that
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}a_kW_k(x)\biggr|\, dx \leqslant\sum_{j=1}^{\mu}\max_{2^{n_{j-1}}\leqslant N<2^{n_j}} \int_0^{1}\biggl|\sum_{k=2^{n_{j-1}}}^{N}a_k^{(j)}W_k(x)\biggr|\, dx<\varepsilon, \\ &\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_ka_kW_k(x)\biggr|\, dx \leqslant\sum_{j=1}^{\mu}\max_{2^{n_{j-1}} \leqslant N<2^{n_j}}\int_0^{1} \biggl| \sum_{k=2^{n_{j-1}}}^{N}\delta_k^{(j)}a_k^{(j)}W_k(x)\biggr|\, dx \\ &\qquad \leqslant A_1\sum_{j=1}^{\mu}|\gamma_j|\,|\Delta_j|=A_1 \int_0^{1}|f_0(x)|\,dx\leqslant A_1\int_0^{1}|f(x)|\,dx. \end{aligned}
\end{equation*}
\notag
$$
Now let us verify condition (7) in Lemma 3. Using (3.62), (3.63), (3.69), (3.72), (3.73), (3.75) and (3.78), for $x\in G$ and $M\in[2^{n_{m-1}},2^{n_{m}})$, $m\in[1,\mu]$, we have
$$
\begin{equation*}
\begin{aligned} \, \biggl|\sum_{k=2^{n_0}}^{M}\delta_ka_kW_k(x)\biggr| &\leqslant\sum_{j=1}^{m-1}|\gamma_j|\chi_{E_j}(x) +\biggl(A_12^{\lambda}|\gamma_{m}|\chi_{G_{m}}(x) +\frac{\varepsilon}{2\mu}\biggr) \,{+}\,\frac{\varepsilon}{2\mu}\chi_{[0,1]\setminus\Delta_{m}}(x) \\ &\leqslant|f(x)|+\frac{A_1|f(x)|}{\delta}+\varepsilon\leqslant\frac{A|f(x)|}{\delta}+\varepsilon, \end{aligned}
\end{equation*}
\notag
$$
where $A=(A_1+1)$. Lemma 3 is proved. § 4. Proof of Theorem 3Let $\varepsilon>0$, and let ($f_{n}(x)\,{\neq}\,0$ for $x \in [0,1)$) be a sequence of polynomials with rational coefficients in the Walsh system. Applying Lemma 3 in succession, we find sequences of sets $\{E_{n}^{(j)}\}_{j=1}^{n}$ and $\{G_{n}^{(j)}\}_{j=1}^{n}$ and polynomials $ \{P_{n}^{(j)}(x)\}_{j=1}^{n}$ and $\{U_{n}^{(j)}(x)\}_{j=1}^{n}$, $n\geqslant1$,
$$
\begin{equation}
U_{n}^{(j)}(x)= \sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1}a_k^{(n,j)}W_k(x), \qquad M_1^{(0)}=2^{m_1^{(0)}}, \quad m_1^{(0)}=2+\biggl[\log_2\frac{1}{\varepsilon}\biggr],
\end{equation}
\tag{4.2}
$$
and
$$
\begin{equation}
P_{n}^{(j)}(x)=\sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1} \delta_k^{(n,j)}a_k^{(n,j)}W_k(x),\delta_k^{(n,j)}=\pm1, \qquad n=1,2,
\end{equation}
\tag{4.3}
$$
where
$$
\begin{equation}
\begin{aligned} \, \notag M_{n}^{(j)}=2^{m_{n}^{(j)}}, \qquad 0&< m_1^{(0)}<m_1^{(1)}=m_{2}^{(0)}<m_{2}^{(1)}<m_{2}^{(2)}<m_{n-1}^{(n-1)}=m_{n}^{(0)} \\ & <m_{n}^{(1)}<\dots<m_{n}^{(n)}=m_{n+1}^{(0)}<m_{n+1}^{(1)}\dots, \end{aligned}
\end{equation}
\tag{4.4}
$$
which for all $1\leqslant j\leqslant n$ satisfy
$$
\begin{equation}
P_{n}^{(j)}(x)=f_{n}(x), \qquad x\in E_{n}^{(j)}, \qquad1\leqslant j\leqslant n,
\end{equation}
\tag{4.5}
$$
$$
\begin{equation}
P_{n}^{(j)}(x)\chi_{[ I_{n}^{(j)},1]}(x)=0, \qquad I_{n}^{(j)}=\frac{1}{M_{n}^{(j)}},
\end{equation}
\tag{4.7}
$$
$$
\begin{equation}
\int_0^{1}|U_{n}^{(j)}(x)|\,dx<\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})} \int_0^{1}\biggl|\sum_{k=M_{n}^{(j-1)}}^{m} a_k^{(n,j)}W_k(x)\biggr|\, dx<2^{-8(n+j)},
\end{equation}
\tag{4.8}
$$
$$
\begin{equation}
\begin{split} \int_0^{1}|P_{n}^{(j)}(x)|\,dx &<\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})}\int_0^{1} \biggl| \sum_{k=M_{n}^{(j-1)}}^{N}\delta_k^{(n,j)}a_k^{(n,j)}W_k(x)\biggr|\, dx \\ &\leqslant A\int_0^{1}|f_{n}(x)|\,dx, \end{split}
\end{equation}
\tag{4.9}
$$
$$
\begin{equation}
\begin{split} &\frac{1}{n}|a_{M_{n-1}-1}^{(n-1,n-1)}| ^|a_k^{(n,j)}| >|a_{k+1}^{(n,j)}| >\dots>|a_{M_{n}^{(j)}}^{(n+1,1)}| \\ &\qquad>|a_{l}^{(n+1,j)}| >|a_{l+1}^{(n+1,j)}|> \dots>|a_{M_{n}^{(j)}}^{(n+1,1)}| \\ &\forall\, k\in[ M_{n}^{(j-1)},M_{n}^{(j)}-1), \quad\forall\, l\in[ M_{n}^{(j)},M_{n}^{(j+1)}-1), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant1, \end{split}
\end{equation}
\tag{4.10}
$$
and
$$
\begin{equation}
\begin{split} &\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})}\biggl| \sum_{k=M_{n}^{(j-1)}}^{N}\delta_k^{(n,j)}a_k^{(n,j)}W_k(x)\biggr| \\ &\qquad\qquad \leqslant A3^{j}|f_n(x)|+2^{-n} \quad\text{for } x\in G_n^{(j)}, \qquad 1\leqslant j\leqslant n, \end{split}
\end{equation}
\tag{4.11}
$$
$$
\begin{equation}
| G_{n}^{(j)}|>1-3^{-j}, \qquad 1\leqslant j\leqslant n,
\end{equation}
\tag{4.12}
$$
where $A$ is the constant from Lemma 3. Using (4.2), (4.7) and (4.8) and employing (4.2) and (2.6), for all ${x\!\in\![I_{n}^{(j)}\!+\!2^{-n},1]}$ and $m\in[ M_{n}^{(j-1)},M_{n}^{(j)})$, $1\leqslant j\leqslant n$, where $ n\geqslant 1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl| \sum_{k=M_{n}^{(j-1)}}^{m}a_k^{(n,j)}W_k(x)\biggr| =\bigl| S_{m}(x,U_{n}^{(j)}(x))\bigr| \\ &\qquad <\frac{2}{2^{-n}}\int_0^{1}|U_{n}^{(j)}(t)|\,dt \leqslant2^{-n-j}, \qquad x\in[ I_{n}^{(j)}+2^{-n},1-2^{-n}]. \end{aligned}
\end{equation}
\tag{4.13}
$$
Next, an appeal to (4.7) shows that
$$
\begin{equation}
I_{n}^{(j)}<I_{n}^{(1)} , \qquad1\leqslant j\leqslant n, \qquad n\geqslant1.
\end{equation}
\tag{4.14}
$$
We set
$$
\begin{equation}
U_0(x):=\sum_{k=0}^{M_1^{(0)}-1}W_k(x), \qquad U_{n}(x):=\sum_{j=1}^{n}U_{n}^{(j)}(x)
\end{equation}
\tag{4.15}
$$
and
$$
\begin{equation}
a_k:=a_k^{(n,j)}, \qquad k\in[ M_{n}^{(j-1)},M_{n}^{(j)}), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant1.
\end{equation}
\tag{4.16}
$$
It is clear (see (4.10), (4.15) and (4.16)) that
$$
\begin{equation}
\sum_{n=1}^{\infty}\biggl( \int_0^{1}|U_{n}(x)|\,dx\biggr) \leqslant\sum_{n=1}^{\infty}\sum_{j=1}^{n}\biggl(\int_0^{1}|U_{n}^{(j)}(x)|\,dx\biggr) <\sum_{n=1}^{\infty}\sum_{j=1}^{n}2^{-n-j}<2
\end{equation}
\tag{4.17}
$$
and Consider the function $U(x)$ defined by
$$
\begin{equation}
U(x):=U_0(x)+\sum_{n=1}^{\infty}U_{n}(x)=U_0(x)+\sum_{n=1}^{\infty} \sum_{j=1}^{n}\sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1}a_kW_k(x).
\end{equation}
\tag{4.19}
$$
Using (4.2), (4.7), (4.8) and (4.15)–(4.19), we conclude that (1) $U(x)\in L^{1}[0,1]$ and $U(x)=0$ for $x\in [\varepsilon,1]$, (2) the Fourier–Walsh series of the function $U(x)$ converges in $L^{1}[0,1)$, and therefore (3) the Fourier–Walsh coefficients of $U(x)$ are monotone decreasing, $c_k(U) \searrow 0$, and $c_k(U)>0$, $k=0,1,2,\dots$ . We claim that the function $U(x)$ is universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1)$. Let $f(x)$ be an arbitrary function from $L^{0}[0,1]$. From the sequence of functions (4.1) we choose a function $f_{\nu_1}(x)$, $\nu_1\geqslant2$, such that
$$
\begin{equation*}
\bigl|\{x\in[0,1]\colon |f(x)-U_0(x)-f_{\nu_1}(x)|\leqslant2^{-8}\}\bigr|\geqslant 1-2^{-4} .
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
H_1:=\bigl\{x\in[0,1]\colon |f(x)-U_0(x)-f_{\nu_1}(x)|\leqslant 2^{-8}\bigr\} .
\end{equation*}
\notag
$$
Now from (4.5) and (4.11) we obtain
$$
\begin{equation*}
\begin{gathered} \, |H_1|\geqslant1-{2^{-4}}, \\ \begin{aligned} \, &\biggl|f(x)-U_0(x)-\sum_{j=1}^{\nu_1}U_j(x)+U_{\nu_1}^{(1)}(x)-P_{\nu_1}^{(1)}(x)\biggr| \\ &\qquad\leqslant2^{-4}+\biggl| \sum_{n=1}^{\nu_1}U{_{n}(x)}\biggr| +|U_{\nu_1}^{(1)}(x)|, \qquad x\in E_{\nu_1}^{(1)}\cap H_1, \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\max_{m\in[ M_{\nu_1}^{(0)},M_{\nu_1}^{(1)})} \biggl|\sum_{k=M_{\nu_1}^{(0)}}^{m}\delta_k^{(\nu_1,1)}a_k^{(\nu_1,1)}W_k(x)\biggr| \\ &\qquad <3^{(2+2)}|f_{\nu_1}(x)|dx+2^{-\nu_{2}-1} \quad\forall\, x\in G_{\nu_1}^{(1)}\cap H_1. \end{aligned}
\end{equation*}
\notag
$$
Assume that we have already defined numbers $0=\nu_0<\nu_1<\dots <\nu_{q-1}$, functions $f_{\nu_1}(x),\dots ,f_{\nu_{q-1}}(x)$, polynomials $\{{P}_{\nu_{r}}^{(r)}{(x)}\}_{r=1}^{q-1}$, $\{\{R_n(x)\}_{n=\nu_{r-1}}^{\nu_{r}-1}\}_{r=1}^{q-1}$, and sets $H_1,H_j,\dots,H_{q-1}$ such that
$$
\begin{equation}
\begin{gathered} \begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1} \biggl[ \biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j}U_n(x) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr) \biggr] \biggr\} \biggr| \\ &\qquad \leqslant2^{-4(q+3)} +\biggl| \sum_{n=\nu_{q-1}+1}^{\nu_q}U_n(x)\biggr| +|U_{\nu_q}^{(q)}(x)| \\ &\qquad \leqslant 2^{-2(q+2)} \quad\forall\, x\in E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1}] \end{split} \end{gathered}
\end{equation}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\max_{m\in[ M_{\nu_{q-1}}^{(q-1)},M_{\nu_{q-1}}^{(q-1)})}\biggl| \sum_{k=M_{\nu_q-1}^{(q-1)}}^{m}\delta_k^{(\nu_{q-1},q-1)}a_k^{(\nu_{q-1},q-1)}W_k(x)\biggr| \leqslant3^{q+2}|f_{\nu_{q-1}}(x)| +2^{-q-1} \\ &\qquad \leqslant3^{q+2}|f_{\nu_q}(x)|\leqslant 2^{-q+1} \quad\forall\, x\in G_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap(E_{\nu_{q-2}}^{(q-2)}\cap H_{q-2}\cap[I_{q-2}^{(1)},1)). \end{aligned}
\end{equation*}
\notag
$$
It is easily seen that we can choose a natural number $\nu_q>\nu_{q-1}+1$ (a function $f_{\nu_q}(x)$ from the sequence (4.1)) and a measurable set $ H_q$ so as to have and
$$
\begin{equation}
\begin{gathered} \, \begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1}\biggl[\sum_{n=\nu_{j-1}+1}^{\nu_j-1}U_n(x)) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\}- f_{\nu_q}(x)\biggr| \\ &\qquad \leqslant 2^{-4(q+3)}, \qquad x\in H_q. \end{split} \end{gathered}
\end{equation}
\tag{4.23}
$$
Hence by (4.5) we have
$$
\begin{equation}
\begin{gathered} \, |f_{\nu_q}(x)|\leqslant{2^{-4(q+3)}} +2^{-4(q+1)} \leqslant2^{-4q}, \\ x\in H_q\cap(E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1}]) , \end{gathered}
\end{equation}
\tag{4.24}
$$
and
$$
\begin{equation}
\begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1}\biggl[\sum_{n=\nu_{j-1}+1}^{\nu_j-1} U_{n}(x))-U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\}- P_{\nu_q}^{(q)}(x)\biggr| \\ &\qquad \leqslant2^{-4(q+3)}, \qquad x\in E_{\nu_q}^{(q)}\cap H_q. \end{split}
\end{equation}
\tag{4.25}
$$
$$
\begin{equation}
P_{n}^{(j)}=0 \quad\text{for } x\in[I_q^{(1)},1-2^{-q}]\subset[I_{n}^{(j)},1-2^{-n}), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant\nu_{q-1},
\end{equation}
\tag{4.26}
$$
by (4.25) we have
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q}\biggl[\biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j}U_{n}(x)\biggr) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\} \biggr| \\ &\qquad \leqslant 2^{-4(q+3)} +\biggl| \sum_{n=\nu_{q-1}+1}^{\nu_q}U_n(x)\biggr| +|U_{\nu_q}^{(q)}(x)| \quad \forall\, x\in E_{\nu_q}^{(q)}\cap H_q\cap[I_q^{(1)},1-2^{-q}). \end{aligned}
\end{equation}
\tag{4.27}
$$
Using (4.11) and (4.24), this shows that
$$
\begin{equation}
\begin{gathered} \, \max_{m\in[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)})} \biggl|\sum_{k=M_{\nu_q}^{(q-1)}}^m\delta_k^{(\nu_q,q)}a_k^{(\nu_q,q)}W_k(x)\biggr| \\ \qquad\qquad\qquad\leqslant3^{q+1}|f_{\nu_q}(x)|+2^{-\nu_q} \leqslant3^{q+1}2^{-4q}+2^{-\nu_q}\leqslant2^{-q+2}, \\ x\in G_{\nu_q}^{(q)}\cap H_q\cap(E_{\nu_{q-1}}^{({q-1})}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1})). \end{gathered}
\end{equation}
\tag{4.28}
$$
Thus, we can recursively define integers $ 0=\nu_0<\nu_1<\dots <\nu_{q-1}<\nu_q<\cdots $ ($\nu_q>\nu_{q-1} +1$) and can choose polynomials $\{P_{\nu_q}^{(q)}(x)\}_{q=1}^{\infty}$ and $\{\{U_m(x)\}_{m=\nu_{q-1}}^{\nu_q-1}\}_{q=1}^{\infty}$ and sets $\{G_{\nu_q}^{(q)}\}_{q=1}^{\infty}$, $\{H_q\}_{q=1}^{\infty}$ and $\{E_{\nu_q}^{(q)}\}_{q=1}^{\infty}$ that satisfy conditions (4.24)–(4.28) for all $q>1$. We set
$$
\begin{equation}
\begin{gathered} \, \delta_k:=\delta_k^{(\nu_q,q)}, \qquad k\in[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)}), \\ \delta_k=1, \qquad k\notin[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)}), \qquad q\geqslant 1, \end{gathered}
\end{equation}
\tag{4.29}
$$
and
$$
\begin{equation}
B:=\bigcup_{k=1}^{\infty}\bigcap_{q=k}^{\infty} \bigl(G_{\nu_q}^{(q)}\cap H_q\cap(E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1} )\cap[I_q^{(1)}+2^{-q},1-2^{-q}]\bigr).
\end{equation}
\tag{4.30}
$$
Hence from (4.6), (4.7), (4.12) and (4.22) we obtain It is clear that $\delta_k=\pm1$ (see (4.3)).We claim that the series converges to $f(x)$ on the set $B$ (that is, almost everywhere on $[0,1)$).Let $ x\in B$. Then there exists a natural number $q_{x}>2$ such that (see (4.30)) $x\in G_{\nu_q}^{(q)}\cap H_q\cap(E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1})\cap[I_q^{(1)}+2^{-q},1-2^{-q})$ for all $q\geqslant q_{x}$. Using (4.2), (4.13), (4.15), (4.27)–(4.29), for each natural number $s\in[ M_{\nu_q}^{(0)},M_{\nu_{q+1}}^{(0)})$, where $q>2$, we have
$$
\begin{equation*}
\begin{aligned} \, &\biggl| f(x)-\sum_{k=0}^{s}\delta_kc_k(U)_kW_k(x)\biggr| \\ &\qquad \leqslant\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q} \biggl[ \biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j} U_n(x)\biggr)-U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr] \biggr\} \biggr| \\ &\qquad\qquad +\sum_{n=\nu_{q-1}}^{\nu_q}\sum_{j=1}^{n} \max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})} \biggl| \sum_{k=M_{n}^{(j)}}^{m}a_k^{(n,j)}W_k(x)\biggr| \\ &\qquad\qquad +\max_{m\in[M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)})} \biggl| \sum_{k=M_{\nu_q}^{(q-1)}}^{m}\delta_k^{(\nu_q,q)}a_k^{(\nu_q,q)}W_k(x)\biggr| \leqslant2^{-q}. \end{aligned}
\end{equation*}
\notag
$$
Hence, since $q\to\infty$ as $s\to\infty$, we conclude that the series (4.31) converges to $f(x)$ almost everywhere on $[0,1)$, that is, the function $U(x)$ is universal in the sense of signs with respect to the Walsh system for the class $L^{0}[0,1]$. Theorem 3 is proved. 
Citation in format AMSBIB
\Bibitem{Gri24} |
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