M. G. Grigoryan, “On universal (in the sense of signs) Fourier series with respect to the Walsh system”, Sb. Math., 215:6 (2024), 717

Processing math: 60%

$\begin{equation*} \lim_{k\to\infty}\int_0^{1}|{f}_k(x)-f(x)|^{p}\,dx=0. \end{equation*} \notag$

$\begin{equation} c_k(f):=\int_0^{1}f(x)\varphi_k(x)\,dx, \qquad k\in\mathbb{N} \cup\{0\}, \end{equation} \tag{1.1}$

$\begin{equation} S_{m}(f): =\sum_{k=0}^{m}c_k(f)\varphi_k(x), \qquad m\in\mathbb{N}\cup\{0\}, \end{equation} \tag{1.2}$

Definition 1. Let $\Omega\subset \Lambda\subseteq\mathbb{N}$ . The density of $\Omega$ with respect to $\Lambda$ is defined by

$\begin{equation} \rho(\Omega)_{\Lambda}:=\lim_{n\to\infty}\frac{\#(\Omega\cap(0,n))}{\#(\Lambda\cap(0,n))}. \end{equation} \tag{1.3}$

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

(a) the series $\sum_{k=0}^{\infty}\delta_kc_k(U)\varphi_k(x)$ is universal for $S$ ,
(b) 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 $|c_k(\widetilde{f})|=|c_k(U)|$ , $k=0,1,2,\dots$ .

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 $|c_k(\widetilde{f})|=|c_k(U)|$ , $k=0,1,2,\dots$ .

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.

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 $L^{0}[0,1]$ .

$\begin{equation*} \lim_{m\to\infty}\int_0^{1} \biggl|\sum_{k=0}^{m}\delta_kc_k(U)\varphi_k(x)- g(x)\biggr| \, dx=0 \end{equation*} \notag$

$\begin{equation*} \lim_{m\to\infty}\int_0^{1} \biggl| \sum_{k=0}^{m}\varepsilon_kc_k(U)\varphi_k(x)- 4g(x)\biggr|\, dx =0. \end{equation*} \notag$

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.

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 $|c_k(\widetilde{f})|=|c_k(U)|$ , $k=0,1,2,\dots$ .

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.

$\begin{equation*} \lim_{m\to\infty}\int_{E_{n}} \biggl| \sum_{k=0}^{m}\varepsilon_kc_k(U)W_k(x)- f(x)\biggr| \, dx=0; \end{equation*} \notag$

$\begin{equation*} \lim_{m\to\infty}\int_{E_{n}} \biggl| \sum_{k=0}^{N_{m}}\delta_kc_k(U)W_k(x)- f(x)\biggr| \, dx=0. \end{equation*} \notag$

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 2. Do Theorems 1–5 hold for the Vilenkin system?

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 $L^{p}[0,1]$ , $p\in [0,1)$ ?

$\begin{equation} W_0(x)=1\quad\text{and} \quad W_n(x)=\prod_{s=1}^{k}r_{m_s}(x), \qquad n=\sum_{s=1}^{k}2^{m_s}, \quad m_1>m_{2}>\dots >m_s, \end{equation} \tag{2.1}$

$\begin{equation*} \begin{gathered} \, r_0(x)= \begin{cases} 1, &x\in\biggl[0,\dfrac{1}{2}\biggr), \\ r-1, &x\in\biggl[\dfrac{1}{2},1\biggr), \end{cases} \\ r_0(x+1)=r_0(x)\quad\text{and} \quad r_k(x)=r_0(2^{k}x), \quad k=1,2,\dots\,. \end{gathered} \end{equation*} \notag$

$\begin{equation} \chi_{E}(x)= \begin{cases} 1, &x\in E, \\ 0, &x\notin E, \end{cases} \end{equation} \tag{2.2}$

$\begin{equation} c_k(g)=\int_0^{1}g(x)W_k(x)\,dx \end{equation} \tag{2.3}$

$\begin{equation} S_{m}(x,g) =\sum_{k=0}^{m}c_k(g)W_k(x). \end{equation} \tag{2.4}$

$\begin{equation} |S_{n}(x,g)| <\frac{2}{\delta}\int_{c}^{d}|g(t)|\,dt \quad \forall\, x\notin[ c-\delta,d+\delta], \end{equation} \tag{2.5}$

$\begin{equation} W_{i}(x)W_j(2^{s}x)=W_{j2^{s}+i}(x) \quad\text{for } 0\leqslant i <2^{s} \text{ (see (2.1))}, \end{equation} \tag{2.6}$

$\begin{equation} \biggl|\sum_{k=0}^{n}W_k(x)\biggr|\leqslant\frac{1}{x}, \end{equation} \tag{2.7}$

$\begin{equation} \sum_{k=0}^{2^{m}-1}W_k(x)= \begin{cases} 2^{m}, & x\in[0,2^{-m}), \\ 0, & x\in[2^{-m},1). \end{cases} \end{equation} \tag{2.8}$

$\begin{equation} \int_0^{1}\biggl|\sum_{k=M}^{N}W_k(x)\biggr|\, dx \leqslant\int_0^{2^{-n}}2^n\,dx+2\int_{2^{-n}}^{1}\frac{1}{x}\,dx\leqslant3n, \end{equation} \tag{2.9}$

$\begin{equation} \sum_{k=2^{m}}^{2^{m+1}-1}W_k(x)= \begin{cases} 2^{m}, & x\in[ 0,2^{-m-1}), \\ -2^{m}, & x\in(2^{-m-1},2^{-m}), \\ 0, & x\in(2^{-m},1]. \end{cases} \end{equation} \tag{2.10}$

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

$\begin{equation*} P(x)=\sum_{k=2^{m}}^{2^{m+1}-1}\beta_kW_k(x) \end{equation*} \notag$

such that

$E^{+}$ and

$E^{-}$ are finite unions of dyadic intervals, and

$\begin{equation*} \begin{aligned} \, &(1)\quad |E^{+}|=|E^{-}|=\frac{|\Delta|}{2}, \\ &(2)\quad |\beta_k|=2^{-(m+\sigma)/2}, \qquad k=2^{m},\dots, 2^{m+1}-1, \\ &(3)\quad P(x)=\pm1, \quad x\in E^{\pm}, \quad P(x)=0, \quad x\notin\Delta, \\ &(4)\quad \max_{2^{m}\leqslant M<2^{m+1}}\biggl| \sum_{k=2^{m}}^{M}\beta_kW_k(x)\biggr| <2^{-(m-\sigma)/2}, \quad x\notin\Delta, \\ &(5)\quad \max_{2^{m}\leqslant M<2^{m+1}}\biggl| \sum_{k=2^{m}}^{M}\beta_kW_k(x)\biggr| <A_0, \quad x\in\Delta, \end{aligned} \end{equation*} \notag$

where

$A_0$ is a constant.

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

$\Delta_{i}^{(1)}$ ,

$i\in[1,N_1]$ ,

$\begin{equation} \Delta=\bigcup_{i=1}^{N_1}\Delta_{i}^{(1)}, \end{equation} \tag{3.1}$

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

$\begin{equation} l_1>\frac{4|\gamma|(A_0+1)}{\eta}+4+l \end{equation} \tag{3.3}$

(

$A_0$ 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

$\begin{equation} l_j>\frac{l_{j-1}+m_{j-1}}{2}, \qquad l_0=m_0=l, \end{equation} \tag{3.4}$

$\begin{equation} l_j>l +4j, \qquad m_j>l_j+2l, \end{equation} \tag{3.5}$

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

$\begin{equation} l_s>\frac{m_{s-1}+l_{s-1}}{2}+4 \end{equation} \tag{3.8}$

and

$\begin{equation} N_s=2^{l_s-l-(j-1)}, \end{equation} \tag{3.9}$

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

$\begin{equation} l_s>l_{s-1}+2>l +4s. \end{equation} \tag{3.12}$

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

$\begin{equation} n_s=m_s+l_s-l-(s-1)+1) \end{equation} \tag{3.15}$

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

$\begin{equation} E_s^{(-)}:=\bigl\{x\in E_{s-1}^{(-)} ;\gamma P_s^{(\Diamond )}(x)<0\bigr\} \quad\text{and} \quad E_s^{(+)}:=\bigl\{x\in E_{s-1}^{(-)} ;\gamma P_s^{(\Diamond)}(x)>0\bigr\}. \end{equation} \tag{3.28}$

From (3.23)–(3.28) we obtain

$\begin{equation} P_s^{(\Diamond)}(x)=0 \quad\forall\, x\notin E_{s-1}^{(-)}, \end{equation} \tag{3.29}$

$\begin{equation} P_s^{(\Diamond)}(x)=2^{s-1}\gamma \quad \forall\, x\in E_s^{(+)}, \end{equation} \tag{3.30}$

$\begin{equation} P_s^{(\Diamond)}(x)=-2^{s-1}\gamma \quad\forall\, x\in E_s^{(-)}; \end{equation} \tag{3.31}$

$E_s^{(-)}$ 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

$\begin{equation} Q_s^{(2)}(x)=0 \quad\forall\, x>2^{-n_0} \end{equation} \tag{3.36}$

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

$\begin{equation} U_s(x)=0 \quad\forall\, x\geqslant2^{-n_0}. \end{equation} \tag{3.41}$

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

$\begin{equation} \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x) \end{equation} \tag{3.50}$

(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 $M\in[2^{n_s}+j2^{m_s},2^{n_s}+(j+1)2^{m_s})$ ,

$\begin{equation} \begin{aligned} \, \notag \int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_kb_kW_k(x)\biggr|\, dx &\leqslant\int_0^{1}\biggl| \sum_{j=1}^{s}{Q}_j^{(1)}(x)\biggr|\, dx +\int_0^{1}\biggl| \sum_{j=1}^{s-1}{Q}_j^{(2)}(x)\biggr|\, dx \\ &\qquad+\int_0^{1}\biggl| \sum_{j=1}^{s-1}P_j^{(\Diamond)}(x)\biggr|\, dx + \int_0^{1}\biggl| \sum_{k=2^{n_s}}^{2^{n_s}+j2^{m_s}-1}a_kW_k(x)\biggr|\, dx \notag \\ &\qquad+\int_0^{1}\biggl|\sum_{k=2^{n_s}+j2^{m_s}}^{M}a_kW_k(x)\biggr|\, dx. \end{aligned} \end{equation} \tag{3.53}$

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

$\begin{equation*} U(x)=\sum_{k=2^{n_0}}^{2^{n}-1}a_kW_k(x) \quad\textit{and}\quad P(x)=\sum_{k=2^{n_0}}^{2^{n}-1}\delta_ka_kW_k(x), \quad\delta_k =\pm1, \end{equation*} \notag$

in the Walsh system such that

$\begin{equation*} \begin{aligned} \, &(1)\quad |E|>1-\varepsilon-2^{-n_0}, \qquad| G|>1-\delta-2^{-n_0}, \\ &(2)\quad 0<a_{k+1}\leqslant a_k<\varepsilon, \quad k\in[\mathit{2}^{n_0},2^{n}-1), \\ &(3)\quad U(x)\cdot\chi_{[2^{-n_0},1]}(x)=0, \\ &(4)\quad P(x)=f(x)\ \ \textit{for}\ \ x\in E, \\ &(5)\quad \max_{2^{n_0}\leqslant M<2^{n}}\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}\delta_ka_kW_k(x)\biggr|\, dx <A\int_0^{1}|f(x)|\,dx, \\ &(6)\quad \int_0^{1}|U(x)|\,dx\leqslant\max_{2^{n_0} \leqslant M<2^{n}}\int_0^{1}\biggl| \sum_{k=2^{n_0}}^{M}a_kW_k(x)\biggr|\, dx<\varepsilon, \\ &(7)\quad \max_{2^{n_0}\leqslant M<2^{n}}\biggl| \sum_{k=2^{n_0}}^{M}\delta _ka_kW_k(x)\biggr| <\frac{A|f(x)|}{\delta}+\varepsilon \quad \forall\, x\in G, \end{aligned} \end{equation*} \notag$

where

$A$ is a constant.

Proof. Let the numbers

$\nu$ and

$\lambda$ be defined by

$\begin{equation} \nu=2+\biggl[ \log_{2}{\frac{1}{\varepsilon}}\biggr]\quad\text{and} \quad \lambda =2+\biggl[ \log_{2}{\frac{1}{\delta}}\biggr]. \end{equation} \tag{3.61}$

Consider the function

$\begin{equation} f_0(x)=f(x)\chi_{[2^{-n_0},1]}(x). \end{equation} \tag{3.62}$

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

$\begin{equation} f_0(x)=\sum_{j=1}^{\mu}\gamma_j\chi_{\Delta_j}(x), \end{equation} \tag{3.63}$

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

$\begin{equation} \begin{cases} 0<a_{k+1}^{(1)}\leqslant a_k^{(1)}<\varepsilon, & k\in [2^{n_0},2^{n_1}-1), \\ 0<a_{k+1}^{(j)}\leqslant a_k^{(j)}<a_{2^{n_{j-1}}-1}^{(j-1)}, & k\in[2^{n_{j-1}},2^{n_j}-1),\quad j\in[2,\mu], \end{cases} \end{equation} \tag{3.67}$

$\begin{equation} U_j(x)\cdot\chi_{[2^{-n_0},1]}(x)=0, \end{equation} \tag{3.68}$

$\begin{equation} P_j(x)=\begin{cases} \gamma_j, & x\in E_j, \\ 0, & x\in[2^{-n_0},1]\setminus\Delta_j, \end{cases} \quad\text{if}\quad\Delta_j\subset[2^{-n_0},1], \end{equation} \tag{3.69}$

$\begin{equation} \int_0^{1}|U_j(x)|\,dx\leqslant \max_{2^{n_{j-1}}\leqslant M<2^{n_j}} \int_0^{1}\biggl| \sum_{k=2^{n_{j-1}}}^{M}a_k^{(j)}W_k(x)\biggr|\, dx <\frac{\varepsilon}{2\mu} \end{equation} \tag{3.70}$

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

$\begin{equation*} P(x)=f_0(x)=f(x) \quad\text{for } x\in E. \end{equation*} \notag$

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.

$\begin{equation} \{f_{n}(x)\}_{n=1}^{\infty} \end{equation} \tag{4.1}$

$\begin{equation} U_{n}^{(j)}(x)= \sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1}a_k^{(n,j)}W_k(x), \qquad M_1^{(0)}=2^{m_1^{(0)}}, \quad m_1^{(0)}=2+\biggl[\log_2\frac{1}{\varepsilon}\biggr], \end{equation} \tag{4.2}$

$\begin{equation} P_{n}^{(j)}(x)=\sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1} \delta_k^{(n,j)}a_k^{(n,j)}W_k(x),\delta_k^{(n,j)}=\pm1, \qquad n=1,2, \end{equation} \tag{4.3}$

$\begin{equation} \begin{aligned} \, \notag M_{n}^{(j)}=2^{m_{n}^{(j)}}, \qquad 0&< m_1^{(0)}<m_1^{(1)}=m_{2}^{(0)}<m_{2}^{(1)}<m_{2}^{(2)}<m_{n-1}^{(n-1)}=m_{n}^{(0)} \\ & <m_{n}^{(1)}<\dots<m_{n}^{(n)}=m_{n+1}^{(0)}<m_{n+1}^{(1)}\dots, \end{aligned} \end{equation} \tag{4.4}$

$\begin{equation} P_{n}^{(j)}(x)=f_{n}(x), \qquad x\in E_{n}^{(j)}, \qquad1\leqslant j\leqslant n, \end{equation} \tag{4.5}$

$\begin{equation} |E_{n}^{(j)}|>1-2^{-8(n+j)}\varepsilon, \end{equation} \tag{4.6}$

$\begin{equation} P_{n}^{(j)}(x)\chi_{[ I_{n}^{(j)},1]}(x)=0, \qquad I_{n}^{(j)}=\frac{1}{M_{n}^{(j)}}, \end{equation} \tag{4.7}$

$\begin{equation} \int_0^{1}|U_{n}^{(j)}(x)|\,dx<\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})} \int_0^{1}\biggl|\sum_{k=M_{n}^{(j-1)}}^{m} a_k^{(n,j)}W_k(x)\biggr|\, dx<2^{-8(n+j)}, \end{equation} \tag{4.8}$

$\begin{equation} \begin{split} \int_0^{1}|P_{n}^{(j)}(x)|\,dx &<\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})}\int_0^{1} \biggl| \sum_{k=M_{n}^{(j-1)}}^{N}\delta_k^{(n,j)}a_k^{(n,j)}W_k(x)\biggr|\, dx \\ &\leqslant A\int_0^{1}|f_{n}(x)|\,dx, \end{split} \end{equation} \tag{4.9}$

$\begin{equation} \begin{split} &\frac{1}{n}|a_{M_{n-1}-1}^{(n-1,n-1)}| ^|a_k^{(n,j)}| >|a_{k+1}^{(n,j)}| >\dots>|a_{M_{n}^{(j)}}^{(n+1,1)}| \\ &\qquad>|a_{l}^{(n+1,j)}| >|a_{l+1}^{(n+1,j)}|> \dots>|a_{M_{n}^{(j)}}^{(n+1,1)}| \\ &\forall\, k\in[ M_{n}^{(j-1)},M_{n}^{(j)}-1), \quad\forall\, l\in[ M_{n}^{(j)},M_{n}^{(j+1)}-1), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant1, \end{split} \end{equation} \tag{4.10}$

$\begin{equation} \begin{split} &\max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})}\biggl| \sum_{k=M_{n}^{(j-1)}}^{N}\delta_k^{(n,j)}a_k^{(n,j)}W_k(x)\biggr| \\ &\qquad\qquad \leqslant A3^{j}|f_n(x)|+2^{-n} \quad\text{for } x\in G_n^{(j)}, \qquad 1\leqslant j\leqslant n, \end{split} \end{equation} \tag{4.11}$

$\begin{equation} | G_{n}^{(j)}|>1-3^{-j}, \qquad 1\leqslant j\leqslant n, \end{equation} \tag{4.12}$

$\begin{equation} \begin{aligned} \, \notag &\biggl| \sum_{k=M_{n}^{(j-1)}}^{m}a_k^{(n,j)}W_k(x)\biggr| =\bigl| S_{m}(x,U_{n}^{(j)}(x))\bigr| \\ &\qquad <\frac{2}{2^{-n}}\int_0^{1}|U_{n}^{(j)}(t)|\,dt \leqslant2^{-n-j}, \qquad x\in[ I_{n}^{(j)}+2^{-n},1-2^{-n}]. \end{aligned} \end{equation} \tag{4.13}$

$\begin{equation} I_{n}^{(j)}<I_{n}^{(1)} , \qquad1\leqslant j\leqslant n, \qquad n\geqslant1. \end{equation} \tag{4.14}$

$\begin{equation} U_0(x):=\sum_{k=0}^{M_1^{(0)}-1}W_k(x), \qquad U_{n}(x):=\sum_{j=1}^{n}U_{n}^{(j)}(x) \end{equation} \tag{4.15}$

$\begin{equation} a_k:=a_k^{(n,j)}, \qquad k\in[ M_{n}^{(j-1)},M_{n}^{(j)}), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant1. \end{equation} \tag{4.16}$

$\begin{equation} \sum_{n=1}^{\infty}\biggl( \int_0^{1}|U_{n}(x)|\,dx\biggr) \leqslant\sum_{n=1}^{\infty}\sum_{j=1}^{n}\biggl(\int_0^{1}|U_{n}^{(j)}(x)|\,dx\biggr) <\sum_{n=1}^{\infty}\sum_{j=1}^{n}2^{-n-j}<2 \end{equation} \tag{4.17}$

$\begin{equation} \{a_k\}_{k=1}^{\infty}\searrow0. \end{equation} \tag{4.18}$

$\begin{equation} U(x):=U_0(x)+\sum_{n=1}^{\infty}U_{n}(x)=U_0(x)+\sum_{n=1}^{\infty} \sum_{j=1}^{n}\sum_{k=M_{n}^{(j-1)}}^{M_{n}^{(j)}-1}a_kW_k(x). \end{equation} \tag{4.19}$

$\begin{equation} a_k=c_k(U), \qquad k=1,2,\dots, \end{equation} \tag{4.20}$

$\begin{equation*} \bigl|\{x\in[0,1]\colon |f(x)-U_0(x)-f_{\nu_1}(x)|\leqslant2^{-8}\}\bigr|\geqslant 1-2^{-4} . \end{equation*} \notag$

$\begin{equation*} H_1:=\bigl\{x\in[0,1]\colon |f(x)-U_0(x)-f_{\nu_1}(x)|\leqslant 2^{-8}\bigr\} . \end{equation*} \notag$

$\begin{equation*} \begin{gathered} \, |H_1|\geqslant1-{2^{-4}}, \\ \begin{aligned} \, &\biggl|f(x)-U_0(x)-\sum_{j=1}^{\nu_1}U_j(x)+U_{\nu_1}^{(1)}(x)-P_{\nu_1}^{(1)}(x)\biggr| \\ &\qquad\leqslant2^{-4}+\biggl| \sum_{n=1}^{\nu_1}U{_{n}(x)}\biggr| +|U_{\nu_1}^{(1)}(x)|, \qquad x\in E_{\nu_1}^{(1)}\cap H_1, \end{aligned} \end{gathered} \end{equation*} \notag$

$\begin{equation*} \begin{aligned} \, &\max_{m\in[ M_{\nu_1}^{(0)},M_{\nu_1}^{(1)})} \biggl|\sum_{k=M_{\nu_1}^{(0)}}^{m}\delta_k^{(\nu_1,1)}a_k^{(\nu_1,1)}W_k(x)\biggr| \\ &\qquad <3^{(2+2)}|f_{\nu_1}(x)|dx+2^{-\nu_{2}-1} \quad\forall\, x\in G_{\nu_1}^{(1)}\cap H_1. \end{aligned} \end{equation*} \notag$

$\begin{equation} |H_{q-1}|>1-2^{-2(q+1)}, \end{equation} \tag{4.21}$

$\begin{equation} \begin{gathered} \begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1} \biggl[ \biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j}U_n(x) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr) \biggr] \biggr\} \biggr| \\ &\qquad \leqslant2^{-4(q+3)} +\biggl| \sum_{n=\nu_{q-1}+1}^{\nu_q}U_n(x)\biggr| +|U_{\nu_q}^{(q)}(x)| \\ &\qquad \leqslant 2^{-2(q+2)} \quad\forall\, x\in E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1}] \end{split} \end{gathered} \end{equation} \notag$

$\begin{equation*} \begin{aligned} \, &\max_{m\in[ M_{\nu_{q-1}}^{(q-1)},M_{\nu_{q-1}}^{(q-1)})}\biggl| \sum_{k=M_{\nu_q-1}^{(q-1)}}^{m}\delta_k^{(\nu_{q-1},q-1)}a_k^{(\nu_{q-1},q-1)}W_k(x)\biggr| \leqslant3^{q+2}|f_{\nu_{q-1}}(x)| +2^{-q-1} \\ &\qquad \leqslant3^{q+2}|f_{\nu_q}(x)|\leqslant 2^{-q+1} \quad\forall\, x\in G_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap(E_{\nu_{q-2}}^{(q-2)}\cap H_{q-2}\cap[I_{q-2}^{(1)},1)). \end{aligned} \end{equation*} \notag$

$\begin{equation} |H_q|>1-2^{-2(q+1)} \end{equation} \tag{4.22}$

$\begin{equation} \begin{gathered} \, \begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1}\biggl[\sum_{n=\nu_{j-1}+1}^{\nu_j-1}U_n(x)) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\}- f_{\nu_q}(x)\biggr| \\ &\qquad \leqslant 2^{-4(q+3)}, \qquad x\in H_q. \end{split} \end{gathered} \end{equation} \tag{4.23}$

$\begin{equation} \begin{gathered} \, |f_{\nu_q}(x)|\leqslant{2^{-4(q+3)}} +2^{-4(q+1)} \leqslant2^{-4q}, \\ x\in H_q\cap(E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1}]) , \end{gathered} \end{equation} \tag{4.24}$

$\begin{equation} \begin{split} &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q-1}\biggl[\sum_{n=\nu_{j-1}+1}^{\nu_j-1} U_{n}(x))-U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\}- P_{\nu_q}^{(q)}(x)\biggr| \\ &\qquad \leqslant2^{-4(q+3)}, \qquad x\in E_{\nu_q}^{(q)}\cap H_q. \end{split} \end{equation} \tag{4.25}$

$\begin{equation} P_{n}^{(j)}=0 \quad\text{for } x\in[I_q^{(1)},1-2^{-q}]\subset[I_{n}^{(j)},1-2^{-n}), \qquad 1\leqslant j\leqslant n, \qquad n\geqslant\nu_{q-1}, \end{equation} \tag{4.26}$

$\begin{equation} \begin{aligned} \, \notag &\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q}\biggl[\biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j}U_{n}(x)\biggr) -U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr]\biggr\} \biggr| \\ &\qquad \leqslant 2^{-4(q+3)} +\biggl| \sum_{n=\nu_{q-1}+1}^{\nu_q}U_n(x)\biggr| +|U_{\nu_q}^{(q)}(x)| \quad \forall\, x\in E_{\nu_q}^{(q)}\cap H_q\cap[I_q^{(1)},1-2^{-q}). \end{aligned} \end{equation} \tag{4.27}$

$\begin{equation} \begin{gathered} \, \max_{m\in[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)})} \biggl|\sum_{k=M_{\nu_q}^{(q-1)}}^m\delta_k^{(\nu_q,q)}a_k^{(\nu_q,q)}W_k(x)\biggr| \\ \qquad\qquad\qquad\leqslant3^{q+1}|f_{\nu_q}(x)|+2^{-\nu_q} \leqslant3^{q+1}2^{-4q}+2^{-\nu_q}\leqslant2^{-q+2}, \\ x\in G_{\nu_q}^{(q)}\cap H_q\cap(E_{\nu_{q-1}}^{({q-1})}\cap H_{q-1}\cap[I_{q-1}^{(1)},1-2^{-q-1})). \end{gathered} \end{equation} \tag{4.28}$

$\begin{equation} \begin{gathered} \, \delta_k:=\delta_k^{(\nu_q,q)}, \qquad k\in[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)}), \\ \delta_k=1, \qquad k\notin[ M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)}), \qquad q\geqslant 1, \end{gathered} \end{equation} \tag{4.29}$

$\begin{equation} B:=\bigcup_{k=1}^{\infty}\bigcap_{q=k}^{\infty} \bigl(G_{\nu_q}^{(q)}\cap H_q\cap(E_{\nu_{q-1}}^{(q-1)}\cap H_{q-1} )\cap[I_q^{(1)}+2^{-q},1-2^{-q}]\bigr). \end{equation} \tag{4.30}$

$\begin{equation*} |B|=1, \qquad \delta_k=\pm1. \end{equation*} \notag$

$\begin{equation} \sum_{k=0}^{\infty}\delta_kc_k(U)_kW_k(x) \end{equation} \tag{4.31}$

$\begin{equation*} \begin{aligned} \, &\biggl| f(x)-\sum_{k=0}^{s}\delta_kc_k(U)_kW_k(x)\biggr| \\ &\qquad \leqslant\biggl| f(x)-\biggl\{U_0(x)+\sum_{j=1}^{q} \biggl[ \biggl(\sum_{n=\nu_{j-1}+1}^{\nu_j} U_n(x)\biggr)-U_{\nu_j}^{(j)}(x)+P_{\nu_j}^{(j)}(x)\biggr] \biggr\} \biggr| \\ &\qquad\qquad +\sum_{n=\nu_{q-1}}^{\nu_q}\sum_{j=1}^{n} \max_{m\in[ M_{n}^{(j-1)},M_{n}^{(j)})} \biggl| \sum_{k=M_{n}^{(j)}}^{m}a_k^{(n,j)}W_k(x)\biggr| \\ &\qquad\qquad +\max_{m\in[M_{\nu_q}^{(q-1)},M_{\nu_q}^{(q)})} \biggl| \sum_{k=M_{\nu_q}^{(q-1)}}^{m}\delta_k^{(\nu_q,q)}a_k^{(\nu_q,q)}W_k(x)\biggr| \leqslant2^{-q}. \end{aligned} \end{equation*} \notag$

\Bibitem{Gri24}

\by M.~G.~Grigoryan

\paper On universal (in the sense of signs) Fourier series with respect to the Walsh system

\jour Sb. Math.

\yr 2024

\vol 215

\issue 6

\pages 717--742

\mathnet{http://mi.mathnet.ru/eng/sm10014}

\crossref{https://doi.org/10.4213/sm10014e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4804035}

\zmath{https://zbmath.org/?q=an:07945692}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215..717G}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001334620600001}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85206873347}

§ 1. Introduction

§ 2. Auxiliary results

§ 3. Proofs of the main lemmas

§ 4. Proof of Theorem 3

Bibliography