Matrix Summability of Walsh–Fourier Series

Abstract

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in $L_1$ space and in $C_W$ space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator $t^{\ast }\left(f\right)$ of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms $t_n\left(f\right)$ of the Walsh–Fourier series are convergent almost everywhere to the function f. The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters.

1. Introduction

The issues of summability of Fourier series have been studied by many authors. In particular, different methods of summabilities are known in the literature. The summability methods are concerned with matrix transformations of partial sums of Walsh–Fourier series. It is well known that the partial sums of Walsh–Fourier series are not convergent in the norm both in the classes of continuous functions and in classes of integrable functions [1] (Chapter 4). It is also known that there is an integral function whose Walsh–Fourier series is divergent at all points [1,2].

An example of matrix transformation is the Fejér or arithmetic mean. In this case, there is a matrix transformation where the elements $\left(t_{k,n}=1/n,1\le k\le n\right)$ of each row of the corresponding triangular matrix are constants. As a result of such a transformation, we obtain a new sequence that can be convergent in the space $C_W$ and $L_1$, and is also convergent almost everywhere for all integrable functions [1,2].

Another example of matrix summability is summability by the Riesz’s logarithmic method ($t_{k,n}=\frac{1}{klogn}$). The new sequence has “good” properties (convergence in the space $C_W$ and $L_1$ as well as convergence almost everywhere for all integrable functions).

From the above, we can assume that if the matrix transformations whose first n element of the nth row represents a non-increasing sequence, then the new sequence obtained as a result of such a transformation is characterized by “good” properties (see estimation (29), Theorem 5 and Corollary 4).

Examples of matrix transformations whose first n element of the nth row represents an increasing sequence are:

• $\left(C,\alpha \right),\alpha >0$ summability ($t_{k,n}=A_{n-k}^{\alpha -1}/A_n^{\alpha },0\le k\le n$), where

  $$A_n^{\alpha }:=\frac{(1+\alpha )\dots (n+\alpha )}{n!};$$
• Nörlund logarithmic summability ($t_{k,n}=\frac{1}{\left(n-k\right)logn},0\le k<n$);
• Cesàro means with varying parameters ($t_{k,n}=A_{n-k}^{{\alpha }_n-1}/A_n^{{\alpha }_n},0\le k\le n,{\alpha }_n\to 0$ as $n\to \infty$).

In the case for $\left(C,\alpha \right)$ summability ($\alpha >0$), it is known that the new sequence obtained by matrix transformation ($t_{k,n}=A_{n-k}^{\alpha -1}/A_n^{\alpha },0\le k\le n$) has “good” properties [1,2,3]. On the other hand, if ($t_{k,n}=\frac{1}{\left(n-k\right)logn},0\le k<n$) or ($t_{k,n}=A_{n-k}^{{\alpha }_n-1}/A_n^{{\alpha }_n},0\le k\le n,{\alpha }_n\to 0$ as $n\to \infty$), then the new sequences are not characterized by “good” properties [4,5].

Therefore, the sequences obtained by matrix transformations can have “good” or “bad” properties. The article sets out the necessary and sufficient conditions for the sequence obtained as a result of the matrix transformation to be convergence in the space $C_W$ and $L_1$ (see Theorem 3, Corollarys 2 and 3, Theorem 4).

Sufficient conditions have been established for the sequence obtained as a result of the matrix transformation to be almost everywhere convergent (see Theorem 6).

Note that the behavior of the sequences obtained as a result of the matrix transformation depends on two-sided estimations of the integral norm (Lebesgue’s constant) of the corresponding kernel of the matrix transformation (see Theorem 1).

The theorems can be used for various methods of summability. At the end of the article, the theorems are used in the case of Cesàro means with varying parameters; this new result improves the theorem of Gát and Abu Joudeh [6].

2. Definitions

Let $\mathbb{P}$ denote the set of positive integers, $\mathbb{N}:=\mathbb{P}\cup \left\{0\right\}$. By a dyadic interval in $\mathbb{I}:=[0,1)$, we mean one of the form $I\left(l,k\right):=\left[\frac{l}{2^k},\frac{l+1}{2^k}\right)$ for some $k\in \mathbb{N}$, $0\le l<2^k$. Given $k\in \mathbb{N}$ and $x\in \mathbb{I},$ let $I_k\left(x\right)$ denote the dyadic interval of length $2^{-k}$ which contains the point x. We use also the notation $I_n:=I_n\left(0\right)\left(n\in \mathbb{N}\right),{\overline{I}}_k\left(x\right):=\mathbb{I}\setminus I_k\left(x\right)$. Let

$$x=\sum _{n=0}^{\infty }{x}_n{2}^{-\left(n+1\right)}$$

be the dyadic expansion of $x\in \mathbb{I}$, where $x_n=0$ or 1, and if x is a dyadic rational number, we choose the expansion which terminates in $0^{\prime }$s. We also use the following notation

$$I_k\left(x\right)=I_k\left(x_0,x_1,\dots ,x_{k-1}\right).$$

For any given $n\in \mathbb{N}$, it is possible to write n uniquely as

$$n=\sum _{k=0}^{\infty }{\epsilon }_k\left(n\right)2^k,$$

where ${\epsilon }_k\left(n\right)=0$ or 1 for $k\in \mathbb{N}$. This expression will be called the binary expansion of n and the numbers ${\epsilon }_k\left(n\right)$ will be called the binary coefficients of n. Let us denote for $1\le n\in \mathbb{N}$, $\left|n\right|:=max\{j\in \mathbb{N}:{\epsilon }_j\left(n\right)\ne 0\}$, that is $2^{\left|n\right|}\le n<2^{\left|n\right|+1}.$

Let us set the definition of the nth $\left(n\in \mathbb{N}\right)$ Walsh–Paley function at point $x\in \mathbb{I}$ as:

$$w_n\left(x\right)={\left(-1\right)}^{{\displaystyle \sum _{j=0}^{\infty }}{\epsilon }_j\left(n\right)x_j}.$$

Let us denote by ∔ the logical addition on $\mathbb{I}$. That is, for any $x,y\in \mathbb{I}$ and $k,n\in \mathbb{N}$

$$x\dotplus y:=\sum _{n=0}^{\infty }\left|x_n-y_n\right|2^{-\left(n+1\right)}.$$

Let us define the binary operator $\oplus :\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ by

$$k\oplus n=\sum _{i=0}^{\infty }\left|{\epsilon }_i\left(k\right)-{\epsilon }_i\left(n\right)\right|2^i.$$

(1)

It is well known (see [1], p. 5) that

$$w_{m\oplus n}\left(x\right)=w_m\left(x\right)w_n\left(x\right),x\in \mathbb{I}\left(n,m\in \mathbb{N}\right).$$

(2)

The Walsh–Dirichlet kernel is defined by

$$D_n\left(x\right)=\sum _{k=0}^{n-1}{w}_k\left(x\right),D_n^{\ast }:=w_n{D}_n.$$

Recall that [1,2]

$$D_{2^n}\left(x\right)=2^n{\chi }_{I_n}\left(x\right),$$

(3)

where ${\chi }_E$ is the characteristic function of the set E,

$$D_n=w_n\sum _{k=0}^{\infty }{\epsilon }_k\left(n\right)r_k{D}_{2^k},$$

(4)

$$D_{2^n+m}=D_{2^n}+w_{2^n}{D}_m\left(m<2^n\right).$$

(5)

The partial sums of Walsh–Fourier series of a function $f\in L_1\left(\mathbb{I}\right)$ are defined as follows: $S_0\left(f\right)=0$ and

$$S_n\left(f;x\right):=\sum _{k=0}^{n-1}\widehat{f}\left(k\right)w_k\left(x\right)\left(n\in \mathbb{N}\right),$$

where $\widehat{f}\left(k\right)=\underset{\mathbb{I}}{\int }f{w}_k$.

3. Triangular Matrix Transforms

Let $T:=\left(t_{k,n}\right)$ be an infinite triangular matrix satisfying the following conditions:

(a)

$t_{k,n}\ge 0,k,n\in \mathbb{N};$

(b)

$t_{k,n}=0,k>n;$

(c)

${\displaystyle \sum _{k=1}^n}{t}_{k,n}=1.$

We define the nth triangular matrix transform of the Walsh–Fourier series by

$$t_n(f;x):=\sum _{k=1}^n{t}_{k,n}{S}_k(f;x)(n\in \mathbb{P}).$$

(6)

The triangular matrix transform kernels are defined by

$$F_n\left(t\right):=\sum _{k=1}^n{t}_{k,n}{D}_k\left(t\right).$$

We have

$$t_n(f,x)=\left(f\ast F_n\right)\left(x\right)={\int }_{\mathbb{I}}f(x\dotplus t)F_n\left(t\right)d\left(t\right).$$

Let us define the following matrices

$$T:=\left[\begin{array}{c}{t}_1\left(f;x\right)\\ ⋮\\ t_n\left(f,x\right)\\ ⋮\end{array}\right],S:=\left[\begin{array}{c}{S}_1\left(f;x\right)\\ ⋮\\ S_n\left(f;x\right)\\ ⋮\end{array}\right],$$

$$m\left(T\right):=\left[\begin{array}{ccccccc}{t}_{11}& 0& 0& \cdots & 0& 0& \cdots \\ t_{12}& t_{22}& 0& \cdots & 0& 0& \cdots \\ ⋮& ⋮& ⋮& & ⋮& ⋮& \\ t_{1n}& t_{2n}& t_{3n}& \cdots & t_{nn}& 0& \cdots \\ ⋮& ⋮& ⋮& & ⋮& \cdots \end{array}\right].$$

Then, equality (6) can be written as follows

$$T=m\left(T\right)\times S.$$

The Fejér means and kernels are denoted by

$$\sigma =m\left(\sigma \right)\times S,$$

where

$$m\left(\sigma \right):=\left[\begin{array}{ccccccc}1& 0& 0& \cdots & 0& 0& \cdots \\ \frac{1}{2}& \frac{1}{2}& 0& \cdots & 0& 0& \cdots \\ ⋮& ⋮& ⋮& & ⋮& ⋮& \\ \frac{1}{n}& \frac{1}{n}& \frac{1}{n}& \cdots & \frac{1}{n}& 0& \cdots \\ ⋮& ⋮& ⋮& & ⋮& \cdots \end{array}\right],\sigma :=\left[\begin{array}{c}{\sigma }_1\left(f;x\right)\\ ⋮\\ {\sigma }_n\left(f;x\right)\\ ⋮\end{array}\right].$$

It is easily seen that

$${\sigma }_n(f,x):=\frac{1}{n}\sum _{k=1}^n{S}_k(f,x),K_n\left(t\right):=\frac{1}{n}\sum _{k=1}^n{D}_k\left(t\right),$$

$${\sigma }_n(f,x)=\left(f\ast K_n\right)\left(x\right)={\int }_{\mathbb{I}}f(x\dotplus t)K_n\left(t\right)d\left(t\right).$$

It is well known that $L_1$ norms of Fejér kernels are uniformly bounded, that is

$$\parallel K_n{\parallel }_1\le c\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(7)

Yano [7] estimated the value of c, and he gave $c=2$. Recently, in paper [8], it was shown that the exact value of c is $\frac{17}{15}$.

4. Auxiliary Results

This section will mention the definitions and notations from the book [1] (Chapter 3).

For each $n\in \mathbb{N}$, let ${\mathcal{A}}_n$ represent the $\sigma$-algebra generated by the collection of dyadic intervals $\left\{I\left(k,n\right):k=0,1,\dots ,2^n-1\right\}$. Thus, every element of ${\mathcal{A}}_n$ is a finite union of intervals of the form $\left[k{2}^{-n},\left(k+1\right)2^{-n}\right)$ or an empty set.

Let $L\left({\mathcal{A}}_n\right)$ represent the collection of ${\mathcal{A}}_n$-measurable functions on $\mathbb{I}$. By the Paley Lemma [1] (Chapter 1, p. 12), $L\left({\mathcal{A}}_n\right)$ coincides with the collection of Walsh polynomials of order less than $2^n$.

A sequence of functions $\left(f_n:n\in \mathbb{N}\right)$ is called a dyadic martingale if each $f_n$ belongs to $L\left({\mathcal{A}}_n\right)$ and

$$\underset{E}{\int }{f}_{n+1}=\underset{E}{\int }{f}_n\left(E\in {\mathcal{A}}_n,n\in \mathbb{N}\right).$$

Let $\mathbf{A}$ denote the collection of sequences $\mathbf{\beta }:=\left\{{\beta }_n:n\in \mathbb{N}\right\}$ which satisfy ${\beta }_n\in$ $L\left({\mathcal{A}}_n\right)$ for $n\in \mathbb{N}$ and

$$∥\mathbf{\beta }∥:=\underset{n\in \mathbb{N}}{sup}{∥{\beta }_n∥}_{\infty }<\infty .$$

For a given $\beta \in \mathbf{A}$ and $f\in L_1\left(\mathbb{I}\right)$, the martingale transform of f is defined by

$$\mathbf{T}\left(\mathbf{\beta }\right)\left(f\right):=\sum _{n=0}^{\infty }{\beta }_n{\Delta }_{n}f,$$

where ${\Delta }_{n}f:=S_{2^{n+1}}\left(f\right)-S_{2^n}\left(f\right)$ for $n\in \mathbb{N}$. The maximal martingale transform is defined by

$${\mathbf{T}}^{*}\left(\mathbf{\beta }\right)\left(f\right):=\underset{N\in \mathbb{N}}{sup}\left|\sum _{n=0}^N{\beta }_n{\Delta }_{n}f\right|.$$

The next Lemma plays an important role in our paper and methods [1] [page 97].

Lemma 1

(Schipp, Simon, Wade and Pál [1]). Let $f\in L_1\left(\mathbb{I}\right),y>0,$ and $\mathbf{\beta }\in \mathbf{A}$. Then, the operator $T^{*}\left(\mathbf{\beta }\right)$ is of weak type (1,1). That is, there exists an absolute constant C such that

$$y\left|\{x\in \mathbb{I}:T^{*}\left(\mathbf{\beta }\right)\left(f\right)>y\}\right|\le {C\parallel \mathbf{\beta }\parallel \parallel f\parallel }_1.$$

5. Kernel Representation and ${\mathit{L}}_{\mathbf{1}}$-Norm of the Matrix Transform Kernels

First, we start with a useful decomposition of the kernel function $F_n^{\ast }:=w_n{F}_n$. We use the next notation in the proof.

$$T_{n,\left(k\right)}:=\sum _{l=1}^k{t}_{l,n},T_n^{\left(k\right)}:=\sum _{l=k}^n{t}_{l,n}$$

and

$$n^{\left(s\right)}:=\sum _{j=s}^{\infty }{\epsilon }_j\left(n\right)2^j,n_{\left(s\right)}:=\sum _{j=0}^s{\epsilon }_j\left(n\right)2^j.$$

We note that ${\displaystyle \sum _{l=1}^n}{t}_{l,n}=T_{n,\left(n\right)}=T_n^{\left(1\right)}$.

Lemma 2.

Let $0<n\in \mathbb{N}$. Then, the next decomposition of the matrix transform kernel holds:

$$\begin{array}{ccc}\hfill F_n^{\ast }& =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)T_n^{\left(n^{\left(s\right)}\right)}\left(D_{2^{s+1}}-D_{2^s}\right)\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n_{\left(s\right)}}\sum _{k=1}^{2^s-1}{t}_{k+n^{\left(s+1\right)},n}{D}_k.\hfill \end{array}$$

Proof of Lemma 2.

For any positive integer n, we write that

$$\begin{array}{ccc}\hfill F_n& =& \sum _{k=1}^n{t}_{k,n}{D}_k=-\sum _{k=1}^{n-1}{T}_{n,\left(k\right)}{w}_k+D_n{T}_{n,\left(n\right)}\hfill \\ & =& -\sum _{k=1}^{n-1}{T}_{n,\left(k\right)}{w}_k+\left(\sum _{k=1}^{n-1}{w}_k\right)T_{n,\left(n\right)}+T_{n,\left(n\right)}\hfill \\ & =& \sum _{k=1}^{n-1}\left(T_{n,\left(n\right)}-T_{n,\left(k\right)}\right)w_k+T_{n,\left(n\right)}\hfill \\ & =& \sum _{k=1}^{n-1}{T}_n^{\left(k+1\right)}{w}_k+T_{n,\left(n\right)}\hfill \\ & =& \sum _{k=0}^{n-1}{T}_n^{\left(k+1\right)}{w}_k.\hfill \end{array}$$

Then, from (2), we have that

$$\begin{array}{ccc}\hfill F_n& =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{k=n^{\left(s+1\right)}}^{n^{\left(s\right)}-1}{T}_n^{\left(k+1\right)}{w}_k\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{k=0}^{2^s-1}{T}_n^{\left(k+1+n^{\left(s+1\right)}\right)}{w}_{k+n^{\left(s+1\right)}}\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n^{\left(s+1\right)}}\sum _{k=0}^{2^s-1}{T}_n^{\left(k+1+n^{\left(s+1\right)}\right)}{w}_k\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n^{\left(s+1\right)}}\sum _{k=1}^{2^s-1}\left(T_n^{\left(k+n^{\left(s+1\right)}\right)}-T_n^{\left(k+1+n^{\left(s+1\right)}\right)}\right)D_k\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n^{\left(s+1\right)}}{T}_n^{\left(n^{\left(s\right)}\right)}{D}_{2^s}.\hfill \end{array}$$

For $x\in I_s$, we have

$$w_{n_{\left(s\right)}}\left(x\right)=w_{2^s}\left(x\right)$$

(8)

Hence,

$$\begin{array}{ccc}\hfill w_n{F}_n& =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)T_n^{\left(n^{\left(s\right)}\right)}{w}_{2^s}{D}_{2^s}\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n_{\left(s\right)}}\sum _{k=1}^{2^s-1}\left(T_n^{\left(k+n^{\left(s+1\right)}\right)}-T_n^{\left(k+1+n^{\left(s+1\right)}\right)}\right)D_k\hfill \\ & =:& F_{n,1}^{\ast }+F_{n,2}^{\ast }.\hfill \end{array}$$

(9)

This completes the proof of Lemma 2. □

We introduce the notation

$$t_n^{*}\left(f\right):=f\ast F_n^{*},t_{n,1}^{*}:=f\ast F_{n,1}^{*},t_{n,2}^{*}:=f\ast F_{n,2}^{*}.$$

Before we discuss the $L_1$-norm of the kernels $F_n$, we prove the following lemma.

Lemma 3.

Let $\left({\alpha }_j:j\in \mathbb{N}\right)$ be a non-decreasing (in sign ${\alpha }_j\uparrow$) bounded sequence of positive real numbers $\mathbf{\alpha }\left(n\right):=({\alpha }_j\left(n\right):={\alpha }_j{\epsilon }_j\left(n\right):j\in \mathbb{N})$. Let the kernel of martingale transform $T\left(\mathbf{\alpha }\left(n\right)\right)f=f\ast M\left(\mathbf{\alpha }\left(n\right)\right)$ be defined by

$$M\left(\mathbf{\alpha }\left(n\right)\right):=\sum _{j=1}^{\infty }{\epsilon }_j\left(n\right){\alpha }_j\left(D_{2^{j+1}}-D_{2^j}\right).$$

Then

$$\parallel M\left(\mathbf{\alpha }\left(n\right)\right){\parallel }_1\sim \sum _{k=1}^{\left|n\right|}|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)|{\alpha }_k.$$

(10)

Proof of Lemma 3.

We write that

$$\begin{array}{ccc}\hfill M\left(\mathbf{\alpha }\left(n\right)\right)& =& \sum _{j=1}^{\left|n\right|-1}({\epsilon }_j\left(n\right){\alpha }_j-{\epsilon }_{j+1}\left(n\right){\alpha }_{j+1})D_{2^{j+1}}\hfill \\ & & +{\epsilon }_{\left|n\right|}\left(n\right){\alpha }_{\left|n\right|}{D}_{2^{\left|n\right|+1}}-{\epsilon }_1\left(n\right){\alpha }_1{D}_{2^1}.\hfill \end{array}$$

This and equality (3) yield that

$$\begin{array}{ccc}& & \parallel M\left(\mathbf{\alpha }\left(n\right)\right){\parallel }_1\le 2\parallel \mathbf{\alpha }\parallel +\sum _{j=1}^{\left|n\right|-1}|{\epsilon }_j\left(n\right){\alpha }_j-{\epsilon }_{j+1}\left(n\right){\alpha }_{j+1}|\hfill \\ & \le & 2\parallel \mathbf{\alpha }\parallel +\sum _{j=1}^{\left|n\right|-1}|{\epsilon }_j\left(n\right)-{\epsilon }_{j+1}\left(n\right)|{\alpha }_j+\sum _{j=1}^{\left|n\right|-1}{\epsilon }_{j+1}\left(n\right)|{\alpha }_j-{\alpha }_{j+1}|.\hfill \end{array}$$

(11)

Since $\mathbf{\alpha }:=({\alpha }_n:n\in \mathbb{N})$ is non-decreasing, we can write

$$\sum _{j=2}^{\left|n\right|-1}{\epsilon }_{j+1}\left(n\right)|{\alpha }_j-{\alpha }_{j+1}|\le \sum _{j=1}^{\left|n\right|-1}|{\alpha }_j-{\alpha }_{j+1}|={\alpha }_{\left|n\right|}-{\alpha }_1\le \parallel \mathbf{f}\mathbf{f}\parallel .$$

(12)

This yields

$$\parallel M_n{\left(\mathbf{\alpha }\right)\parallel }_1\le 3∥\mathbf{\alpha }∥+\sum _{j=2}^{\left|n\right|-1}|{\epsilon }_j\left(n\right)-{\epsilon }_{j+1}\left(n\right)|{\alpha }_j.$$

(13)

Now, we show the lower estimate for $\parallel M_n{\left(\mathbf{\alpha }\right)\parallel }_1$. We use the construction in the book ([1], p. 35). Let us choose the strictly monotone increasing sequences $a_i$ and $b_i$ ($i=1,\dots ,s$) such that

$$0<a_1\le b_1<a_2\le b_2<\dots <a_s\le b_s<a_{s+1}=\infty .$$

It is easy to see that

$$b_j+1<a_{j+1}$$

holds. We define the nature number $n={\sum }_{j=0}^{\infty }{\epsilon }_j\left(n\right)2^j$ by

$${\epsilon }_j\left(n\right):=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{a}_i\le j\le b_i\mathrm{for}\mathrm{an}i\in \{1,\dots ,s\},\hfill \\ 0,\hfill & \mathrm{if}{b}_i<j<a_{i+1}\mathrm{for}\mathrm{an}i\in \{1,\dots ,s\}\mathrm{or}j<a_1.\hfill \end{array}\right.$$

(14)

Let us set the sets

$$A_k:=\left(\frac{1}{2^{a_k+1}},\frac{1}{2^{a_k}}\right),B_k:=\left(\frac{1}{2^{b_k+2}},\frac{1}{2^{b_k+1}}\right),k=1,\dots ,s.$$

For $x\in A_k$, we have that

$$\begin{array}{ccc}& & |M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)|=\left|\sum _{j=1}^{\left|n\right|}{\epsilon }_j\left(n\right){\alpha }_j(D_{2^{j+1}}\left(x\right)-D_{2^j}\left(x\right))\right|\hfill \\ & =& \left|\sum _{i=1}^{k-1}\sum _{j=a_i}^{b_i}{\alpha }_j(D_{2^{j+1}}\left(x\right)-D_{2^j}\left(x\right))+\sum _{j=a_k}^{b_k}{\alpha }_j(D_{2^{j+1}}\left(x\right)-D_{2^j}\left(x\right))\right|\hfill \\ & =& \left|\sum _{i=1}^{k-1}\sum _{j=a_i}^{b_i}{\alpha }_j{2}^j-{\alpha }_{a_k}{2}^{a_k}\right|.\hfill \end{array}$$

The construction of the sequences $\left\{a_k\right\}$ and $\left\{b_k\right\}$ yields

$$\begin{array}{ccc}\hfill \sum _{i=1}^{k-1}\sum _{j=a_i}^{b_i}{\alpha }_j{2}^j& \le & {\alpha }_{b_{k-1}}\sum _{i=1}^{k-1}(2^{b_i+1}-2^{a_i})\hfill \\ & \le & {\alpha }_{b_{k-1}}\sum _{i=1}^{k-1}(2^{b_i+1}-2^{b_{i-1}+1})\hfill \\ & \le & {\alpha }_{b_{k-1}}{2}^{b_{k-1}+1}\le {\alpha }_{a_k}{2}^{b_{k-1}+1}\hfill \end{array}$$

and

$$|M_n\left(\mathbf{\alpha }\right)\left(x\right)|\ge {\alpha }_{a_k}{2}^{a_k}-{\alpha }_{a_k}{2}^{b_{k-1}+1}\ge {\alpha }_{a_k}{2}^{a_k-1}.$$

That is, we obtain that

$${\int }_{A_k}\left|M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)\right|dx\ge {\alpha }_{a_k}{2}^{a_k-1}{2}^{-a_k-1}\ge \frac{{\alpha }_{a_k-1}}{4}.$$

(15)

Now, we set $x\in B_k$.

$$\begin{array}{ccc}\hfill |M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)|& =& \left|\sum _{i=1}^k\sum _{j=a_i}^{b_i}{\alpha }_j(D_{2^{j+1}}\left(x\right)-D_{2^j}\left(x\right))\right|\hfill \\ & =& \sum _{i=1}^k\sum _{j=a_i}^{b_i}{\alpha }_j{2}^j\ge {\alpha }_{b_k}{2}^{b_k}\hfill \end{array}$$

and

$${\int }_{B_k}\left|M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)\right|dx\ge {\alpha }_{b_k}{2}^{b_k}{2}^{-b_k-2}=\frac{{\alpha }_{b_k}}{4}.$$

(16)

The sets $A_k$ and $B_k$ are pairwise disjoint intervals ($k=1,\dots ,s$), and we have

$$\begin{array}{ccc}\hfill \parallel M\left(\mathbf{\alpha }\left(n\right)\right){\parallel }_1& \ge & \sum _{k=1}^s\left({\int }_{A_k}|M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)|dx+{\int }_{B_k}\left|M\left(\mathbf{\alpha }\left(n\right)\right)\left(x\right)\right|dx\right)\hfill \\ & \ge & \frac{1}{4}\sum _{k=1}^s({\alpha }_{a_k-1}+{\alpha }_{b_k})\hfill \end{array}$$

(see inequalities (15) and (16) as well). Taking into account that

$$|{\epsilon }_j\left(n\right)-{\epsilon }_{j+1}\left(n\right)|=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}j=a_k-1\mathrm{or}j=b_k\mathrm{for}\mathrm{a}k\in \{1,\dots ,s\},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

we conclude that

$$\parallel M\left(\mathbf{\alpha }\left(n\right)\right){\parallel }_1\ge \frac{1}{4}\sum _{j=1}^{\left|n\right|}|{\epsilon }_j\left(n\right)-{\epsilon }_{j+1}\left(n\right)|{\alpha }_j.$$

(17)

Summarizing our results in inequalities (13) and (17), we complete the proof. □

Theorem 1.

(a) If the sequence $\{t_{k,n}:1\le k\le n\}$ is monotone non-increasing (in sign $t_{k,n}\downarrow$) for any fixed n, then there exists a positive constant c such that

$$\parallel F_n{\parallel }_1\le c$$

(18)

holds for all $n\in \mathbb{P}$.

(b) If the sequence $\{t_{k,n}:1\le k\le n\}$ is monotone non-decreasing (in sign $t_{k,n}\uparrow$) for any fixed n, then

$$\parallel F_n{\parallel }_1\sim \sum _{s=1}^{\left|n\right|}|{\epsilon }_s\left(n\right)-{\epsilon }_{s+1}\left(n\right)|T_n^{\left(n^{\left(s\right)}\right)}.$$

(19)

Proof of Theorem 1.

First, let the sequence $\{t_{k,n}:1\le k\le n\}$ be monotone non-increasing (in sign $t_{k,n}\downarrow$). For the kernel $F_n$, we apply Abel’s transformation

$$F_n=\sum _{k=1}^{n-1}(t_{k,n}-t_{k+1,n})k{K}_k+t_{n,n}n{K}_n.$$

(20)

Inequality (7) implies that

$$\begin{array}{ccc}\hfill \parallel F_n{\parallel }_1& \le & \sum _{k=1}^{n-1}|t_{k,n}-t_{k+1,n}|k\parallel K_k{\parallel }_1+t_{n,n}n{\parallel K_n\parallel }_1\hfill \\ & \le & c\left(\sum _{k=1}^{n-1}(t_{k,n}-t_{k+1,n})k+t_{n,n}n\right)\hfill \\ & \le & c\sum _{k=1}^n{t}_{k,n}\le c.\hfill \end{array}$$

(21)

Second, let the sequence $\{t_{k,n}:1\le k\le n\}$ be monotone non-decreasing (in sign $t_{k,n}\uparrow$). Theorem 2 yields that

$$\parallel F_n{\parallel }_1=\parallel F_n^{\ast }{\parallel }_1\le \parallel F_{n,1}^{\ast }{\parallel }_1+{\parallel F_{n,2}^{\ast }\parallel }_1.$$

Applying Lemma 3 with setting ${\alpha }_s:=T_n^{\left(n^{\left(s\right)}\right)}$, we obtain

$$\parallel F_{n,1}^{\ast }{\parallel }_1\sim \sum _{s=1}^{\left|n\right|}|{\epsilon }_s\left(n\right)-{\epsilon }_{s+1}\left(n\right)|T_n^{\left(n^{\left(s\right)}\right)}.$$

At last, we discuss the norm $\parallel F_{n,2}^{\ast }{\parallel }_1$. In case ${\epsilon }_s\left(n\right)=1$, we write that

$$\begin{array}{ccc}\hfill I_s:=& & \sum _{k=1}^{2^s-1}{t}_{k+n^{\left(s+1\right)},n}{D}_k=\sum _{k=1}^{2^s-1}{t}_{k+n^{\left(s\right)}-2^s,n}{D}_k\hfill \\ & =& \sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}{D}_{2^s-l}(s=0,\dots ,|n|-1).\hfill \end{array}$$

(22)

For $s=\left|n\right|$, we have that

$$I_{\left|n\right|}:=\sum _{k=1}^{2^{\left|n\right|}-1}{t}_{k,n}{D}_k=\sum _{l=1}^{2^{\left|n\right|}-1}{t}_{2^{\left|n\right|}-l,n}{D}_{2^{\left|n\right|}-l}.$$

It is known that

$$D_{2^k-j}=D_{2^k}-w_{2^k-1}{D}_j\mathrm{for}j=1,\dots ,2^k-1.$$

(23)

Applying equality (23) and Abel’s transformation, we obtain

$$\begin{array}{ccc}\hfill I_s& =& D_{2^s}\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}-w_{2^s-1}\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}{D}_l\hfill \\ & =& D_{2^s}\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}\hfill \\ & & -w_{2^s-1}\left(\sum _{l=1}^{2^s-2}(t_{n^{\left(s\right)}-l,n}-t_{n^{\left(s\right)}-l-1,n})l{K}_l+t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)K_{2^s-1}\right).\hfill \end{array}$$

(24)

Analogously, we transform the expression $I_{\left|n\right|}$. Inequality (7) yields

$$\parallel I_s{\parallel }_1\le c\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}(s=0,\dots ,|n\left|\right)$$

and

$$\parallel I_{\left|n\right|}{\parallel }_1\le \sum _{l=1}^{2^{\left|n\right|}-1}{t}_{2^{\left|n\right|}-l,n}.$$

Thus,

$$\parallel F_{n,2}^{\ast }{\parallel }_1={∥\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n_{\left(s\right)}}{I}_s∥}_1\le \sum _{s=0}^{\left|n\right|}{\parallel I_s\parallel }_1\le c\sum _{k=1}^n{t}_{k,n}\le c.$$

(25)

Theorem 1 is proved. □

6. Convergence in Measure of Matrix Transform of Walsh–Fourier Series

Theorem 2.

Let $\{t_{k,n}:1\le k\le n\}$ be a monotone non-decreasing (or monotone non-increasing) sequence for any fixed n. Then, there exists a positive constant c such that

$$y\left|\{x\in \mathbb{I}:|t_n\left(f\right)|>y\}\right|\le c{\parallel f\parallel }_1$$

holds for all $f\in L^1\left(\mathbb{I}\right)$ and $y>0$.

Proof of Theorem 2.

First, let the sequence $\{t_{k,n}:1\le k\le n\}$ be monotone non-increasing (in sign $t_{k,n}\downarrow$). Since, by Theorem 1, we write that

$$\parallel t_n{\left(f\right)\parallel }_1=\parallel f\ast F_n{\parallel }_1\le {\parallel f\parallel }_1\parallel F_n{\parallel }_1\le c{\parallel f\parallel }_1.$$

(26)

(for more details, see [1,2]). We immediately learn that the operator $t_n$ is of weak type (1,1).

Second, let the sequence $\{t_{k,n}:1\le k\le n\}$ be monotone non-decreasing (in sign $t_{k,n}\uparrow$). Lemma 2 yields that

$$t_n^{*}\left(f\right)=f\ast F_n^{*}=f\ast F_{n,1}^{*}+f\ast F_{n,2}^{*}.$$

Since $t_{n,1}^{*}\left(f\right)=f\ast F_{n,1}^{*}$ is a martingale transform with coefficients ${\epsilon }_s\left(n\right)T_n^{\left(n^{\left(s\right)}\right)}$, we apply Lemma 1. This lemma gives immediately that the operator $t_{n,1}^{*}$ is of weak type $(1,1)$. That is, there exists a positive constant c such that

$$y\left|\{x\in \mathbb{I}:|t_{n,1}^{*}\left(f\right)|>y\}\right|\le c{\parallel f\parallel }_1(y>0)$$

(27)

holds for all $f\in L_1\left(\mathbb{I}\right)$.

For the operator $t_{n,2}^{*}$, we apply inequality (25) and write that

$$\parallel t_{n,2}^{*}{\left(f\right)\parallel }_1=\parallel f\ast F_{n,2}{\parallel }_1\le {\parallel f\parallel }_1\parallel F_{n,2}^{*}{\parallel }_1\le c{\parallel f\parallel }_1.$$

(28)

(for more details, see [1,2]). That is, the operator $t_{n,2}^{*}$ is of weak type (1,1).

Inequalities (26)–(28) complete the proof of Theorem 2. □

Theorem 2 implies that the following is valid.

Corollary 1.

Let $\{t_{k,n}:1\le k\le n\}$ be a monotone non-decreasing (or monotone non-increasing) sequence for any fixed n. Then, for all $f\in L_1\left(\mathbb{I}\right),$$t_n\left(f\right)\to f$ in measure as $n\to \infty$.

Remark 1.

In the case that the sequence $\{t_{k,n}:1\le k\le n\}$ is not increasing for any fixed n, below, more is proved. In particular, the weak type inequality for the maximal operator $t^{*}\left(f\right)$ is proved (see Theorem 5).

7. Convergence in ${\mathit{L}}_{\mathbf{1}}$-Norm and ${\mathit{C}}_{\mathit{W}}$-Norm

Let $C_W\left(\mathbb{I}\right)$ represent the collection of functions f which are continuous at every dyadic irrational, continuous from the right on $\mathbb{I}$, and have a finite limit from the left on $\mathbb{I}$, all this in the usual topology.

Set ${\parallel f\parallel }_{C_W}:={sup}_{x\in \mathbb{I}}\left|f\left(x\right)\right|$. Let us denote by $L_p\left(\mathbb{I}\right)$ the usual Lebesgue spaces on $\mathbb{I}$ with the corresponding norm ${\parallel .\parallel }_p$ ($1\le p<\infty$). Let $X:=X\left(\mathbb{I}\right)$ be either $L_1\left(\mathbb{I}\right)$ or $C_W\left(\mathbb{I}\right)$ with the corresponding norm denoted by ${\parallel .\parallel }_X$. The modulus of continuity, when $X=C_W\left(\mathbb{I}\right)$, and the integrated modulus of continuity, while $X=L_1\left(\mathbb{I}\right)$ are defined by

$$\omega {\left(\frac{1}{2^n},f\right)}_X:=\underset{h\in I_n}{sup}{\parallel f(.\dotplus h)-f(.)\parallel }_X.$$

In this section, we discuss the convergence of matrix transforms in $L_1$ space and in $C_W$ in terms of modulus of continuity and matrix transform variation. Moreover, in Theorem 4, we show the sharpness of our result.

For non-negative integer n, the variation of n is defined by

$$V\left(n\right):=\sum _{k=0}^{\infty }|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)|+{\epsilon }_0\left(n\right)$$

(see [1], p. 34). Motivated by this definition for the monotone non-decreasing sequence $\{t_{k,n}:1\le k\le n\}$ (in sign $t_{k,n}\uparrow$), we introduce the matrix transform variation of n by

$$V(n,\left\{t_{k,n}\right\}):=\sum _{k=1}^{\left|n\right|}|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)|T_n^{\left(n^{\left(k\right)}\right)}.$$

For the convenience of the reader, the main theorems of this section will be formulated first, and the proofs will be given below.

Theorem 3.

Let $f\in X\left(\mathbb{I}\right)$ and $\{t_{k,n}:1\le k\le n\}$ be a sequence of non-negative numbers.

(a) If the sequence $\{t_{k,n}:1\le k\le n\}$ is monotone non-increasing (in sign $t_{k,n}\downarrow$), then

$$\begin{array}{ccc}\hfill \parallel t_n{\left(f\right)-f\parallel }_X& \le & c_1\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X+c_2\omega {\left(\frac{1}{2^{\left|n\right|-1}},f\right)}_X\hfill \\ & & +c_3\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^r,n}\omega {\left(\frac{1}{2^r},f\right)}_X.\hfill \end{array}$$

(29)

(b) If the sequence $\{t_{k,n}:1\le k\le n\}$ is monotone non-decreasing (in sign $t_{k,n}\uparrow$), then

$$\begin{array}{ccc}\hfill \parallel t_n{\left(f\right)-f\parallel }_X& \le & c_{1}V(n,\left\{t_{k,n}\right\})\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X\hfill \\ & & +c_2\omega {\left(\frac{1}{2^{\left|n\right|-1}},f\right)}_X\hfill \\ & & +c_3\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^{r+1}-1,n}\omega {\left(\frac{1}{2^r},f\right)}_X.\hfill \end{array}$$

(30)

Proof of Theorem 3.

We carry out the proof of Theorem 3 for space $X=L_1\left(\mathbb{I}\right)$. The proof for $X=C_W$ is similar and even simpler. Keeping in mind that ${\sum }_{k=1}^n{t}_{k,n}=1$, we write that

$$\begin{array}{ccc}\hfill t_n(f,x)-f\left(x\right)& =& {\int }_{\mathbb{I}}(f(x\dotplus t)-f\left(x\right))\sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & & +{\int }_{\mathbb{I}}(f(x\dotplus t)-f\left(x\right))\sum _{k=2^{\left|n\right|-1}}^{2^{\left|n\right|}-1}{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & & +{\int }_{\mathbb{I}}(f(x\dotplus t)-f\left(x\right))\sum _{k=1}^{2^{\left|n\right|-1}-1}{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & =:& I_1+I_2+I_3.\hfill \end{array}$$

(31)

First, we discuss the expression $I_1$. We write that

$$\begin{array}{ccc}\hfill I_1& =& {\int }_{\mathbb{I}}(f(x\dotplus t)-S_{2^{\left|n\right|}}(f,x\dotplus t))\sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & & +{\int }_{\mathbb{I}}(S_{2^{\left|n\right|}}(f,x\dotplus t)-S_{2^{\left|n\right|}}(f,x))\sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & & +{\int }_{\mathbb{I}}(S_{2^{\left|n\right|}}(f,x)-f\left(x\right))\sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k\left(t\right)dt\hfill \\ & =:& I_{1,1}+I_{1,2}+I_{1,3}.\hfill \end{array}$$

(32)

It is easily seen that $I_{1,2}=0$. Applying generalized Minkowski’s inequality, we have

$$\parallel I_{1,1}{\parallel }_X\le \omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X{\int }_{\mathbb{I}}\left|\sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k\left(t\right)\right|dt.$$

(33)

For sequence $t_{k,n}\uparrow$, we learn immediately that

$$\parallel I_{1,1}{\parallel }_X\le \omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X(1+V(n,\left\{t_{k,n}\right\})).$$

Analogously, we can prove that

$$\parallel I_{1,3}{\parallel }_X\le \omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X(1+V(n,\left\{t_{k,n}\right\})).$$

That is, we have that

$$\parallel I_1{\parallel }_X\le \omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X(1+V(n,\left\{t_{k,n}\right\})).$$

(34)

For sequence $t_{k,n}\downarrow$ we apply the equality (5), and we obtain

$$\begin{array}{ccc}\hfill \sum _{k=2^{\left|n\right|}}^n{t}_{k,n}{D}_k& =& \sum _{l=0}^{n-2^{\left|n\right|}}{t}_{2^{\left|n\right|}+l,n}{D}_{2^{\left|n\right|}+l}\hfill \\ & =& \sum _{l=0}^{n-2^{\left|n\right|}}{t}_{2^{\left|n\right|}+l,n}{D}_{2^{\left|n\right|}}+w_{2^{\left|n\right|}}\sum _{l=1}^{n-2^{\left|n\right|}}{t}_{2^{\left|n\right|}+l,n}{D}_l.\hfill \end{array}$$

Applying Abel’s transform and inequalities (7) and (33), we learn that

$$\parallel I_{1,1}{\parallel }_X\le c\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X.$$

Analogously, we can prove that

$$\parallel I_{1,3}{\parallel }_X\le c\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X.$$

That is, we have that

$$\parallel I_1{\parallel }_X\le c\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X.$$

(35)

The estimation of the $I_2$ is analogous to the estimation of the $I_1$, and we have

$$\parallel I_2{\parallel }_X\le c\omega {\left(\frac{1}{2^{\left|n\right|-1}},f\right)}_X{\int }_{\mathbb{I}}\left|\sum _{k=2^{\left|n\right|-1}}^{2^{\left|n\right|}-1}{t}_{k,n}{D}_k\left(t\right)\right|dt.$$

Now, we discuss the integral $I:={\int }_{\mathbb{I}}\left|{\sum }_{k=2^{\left|n\right|-1}}^{2^{\left|n\right|}-1}{t}_{k,n}{D}_k\left(t\right)\right|dt$. We apply equality (23), Abel’s transformation and inequality (7). We have that

$$\begin{array}{ccc}\hfill I& \le & {\int }_{\mathbb{I}}\left|\sum _{k=1}^{2^{\left|n\right|-1}}{t}_{2^{\left|n\right|}-k,n}{D}_{2^{\left|n\right|}-k}\left(t\right)\right|dt\hfill \\ & \le & \sum _{k=1}^{2^{\left|n\right|-1}}{t}_{2^{\left|n\right|}-k,n}+{\int }_{\mathbb{I}}\left|\sum _{k=1}^{2^{\left|n\right|-1}}{t}_{2^{\left|n\right|}-k,n}{D}_k\left(t\right)\right|dt\hfill \\ & \le & c+{\int }_{\mathbb{I}}\left|\sum _{k=1}^{2^{\left|n\right|-1}-1}(t_{2^{\left|n\right|}-k,n}-t_{2^{\left|n\right|}-k-1,n})k{K}_k\left(t\right)\right|dt\hfill \\ & & +{\int }_{\mathbb{I}}{t}_{2^{\left|n\right|-1},n}{2}^{\left|n\right|-1}\left|K_{2^{\left|n\right|-1}}\left(t\right)\right|dt\hfill \\ & \le & c+c\left(\sum _{k=1}^{2^{\left|n\right|-1}-1}\left|t_{2^{\left|n\right|}-k,n}-t_{2^{\left|n\right|}-k-1,n}\right|k+t_{2^{\left|n\right|-1},n}{2}^{\left|n\right|-1}.\right)\hfill \end{array}$$

For sequence $t_{k,n}\uparrow$, we learn that

$$I\le c+c\sum _{l=1}^{2^{\left|n\right|-1}}{t}_{2^{\left|n\right|}-l}\le c.$$

For sequence $t_{k,n}\downarrow$, we write

$$I\le c+c{t}_{2^{\left|n\right|-1},n}{2}^{\left|n\right|-1}\le c+c\sum _{k=0}^{2^{\left|n\right|-1}-1}{t}_{2^{\left|n\right|-1}-k,n}\le c.$$

That is, we have that

$$\parallel I_2{\parallel }_X\le c\omega {\left(\frac{1}{2^{\left|n\right|-1}},f\right)}_X$$

(36)

in both cases (a) and (b).

At last, we discuss the expression $I_3$.

$$\begin{array}{ccc}\hfill I_3& =& \sum _{r=0}^{\left|n\right|-2}\sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{\int }_{\mathbb{I}}(f(x\dotplus t)-f\left(x\right))D_j\left(t\right)dt\hfill \\ & =& \sum _{r=0}^{\left|n\right|-2}{\int }_{\mathbb{I}}(f(x\dotplus t)-S_{2^r}(f,x\dotplus t))\sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j\left(t\right)dt\hfill \\ & & +\sum _{r=0}^{\left|n\right|-2}{\int }_{\mathbb{I}}(S_{2^r}(f,x\dotplus t)-S_{2^r}(f,x))\sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j\left(t\right)dt\hfill \\ & & +\sum _{r=0}^{\left|n\right|-2}{\int }_{\mathbb{I}}(S_{2^r}(f,x)-f\left(x\right))\sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j\left(t\right)dt\hfill \\ & =:& I_{3,1}+I_{3,2}+I_{3,3}.\hfill \end{array}$$

It can be proved that $I_{3,2}=0$. By generalized Minkowski’s inequality, we have that

$$\parallel I_{3,i}{\parallel }_1\le \sum _{r=0}^{\left|n\right|-2}\omega {\left(\frac{1}{2^r},f\right)}_X{\int }_{\mathbb{I}}\left|\sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j\left(t\right)\right|dt(i=1,3).$$

Equality (5) and Abel’s transformation yield that

$$\begin{array}{ccc}& & \sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j=\sum _{j=0}^{2^r-1}{t}_{2^r+j,n}{D}_{2^r}+w_{2^r}\sum _{j=1}^{2^r-1}{t}_{2^r+j,n}{D}_j\hfill \\ & =& \sum _{j=0}^{2^r-1}{t}_{2^r+j,n}{D}_{2^r}\hfill \\ & +& w_{2^r}\left(\sum _{j=1}^{2^r-2}(t_{2^r+j,n}-t_{2^r+j+1,n})j{K}_j+t_{2^{r+1}-1,n}(2^{r+1}-1)K_{2^{r+1}-1}\right).\hfill \end{array}$$

Inequality (7) gives

$$\parallel \sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j{\parallel }_1\le \sum _{j=0}^{2^r-1}{t}_{2^r+j,n}+c\left(\sum _{j=1}^{2^r-2}|t_{2^r+j,n}-t_{2^r+j+1,n}|j+t_{2^{r+1}-1,n}(2^{r+1}-1)\right).$$

For sequence $t_{k,n}\downarrow$, we write

$$\parallel \sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j{\parallel }_1\le c\sum _{j=0}^{2^r-1}{t}_{2^r+j,n}\le c{2}^r{t}_{2^r,n}.$$

For sequence $t_{k,n}\uparrow$, we have

$$\parallel \sum _{j=2^r}^{2^{r+1}-1}{t}_{j,n}{D}_j{\parallel }_1\le c{2}^r{t}_{2^{r+1}-1,n}.$$

That is, for a monotone non-increasing sequence (in sign $t_{k,n}\downarrow$), we have

$$\parallel I_{3,i}{\parallel }_X\le c\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^r,n}\omega {\left(\frac{1}{2^r},f\right)}_X(i=1,3)$$

(37)

and for a monotone non-decreasing sequence (in sign $t_{k,n}\uparrow$),

$$\parallel I_{3,i}{\parallel }_X\le c\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^{r+1}-1,n}\omega {\left(\frac{1}{2^r},f\right)}_X(i=1,3).$$

(38)

For a monotone non-increasing sequence (in sign $t_{k,n}\downarrow$), we proved that

$$\parallel I_3{\parallel }_X\le c\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^r,n}\omega {\left(\frac{1}{2^r},f\right)}_X.$$

(39)

For a monotone non-decreasing sequence (in sign $t_{k,n}\uparrow$), we reached that

$$\parallel I_3{\parallel }_X\le c\sum _{r=0}^{\left|n\right|-2}{2}^r{t}_{2^{r+1}-1,n}\omega {\left(\frac{1}{2^r},f\right)}_X.$$

(40)

Combining (31), (34)–(36), (39) and (40), we complete the proof. □

Corollary 2.

Let $f\in X\left(\mathbb{I}\right)$ and $\{m_n:n\in \mathbb{P}\}$ be a strictly monotone increasing sequence. Let $\{t_{l,m_n}:1\le l\le m_n\}$ be a monotone non-decreasing sequence of non-negative numbers (in sign $t_{l,m_n}\uparrow$). Let the condition

$$\omega {\left(\frac{1}{2^{|m_n|}},f\right)}_X=o\left(\frac{1}{V(m_n,\left\{t_{l,m_n}\right\})}\right)$$

(41)

be satisfied. Then, the subsequence $t_{m_n}\left(f\right)$ converges in the norm of the space $X\left(\mathbb{I}\right)$.

Corollary 3.

Let $f\in X\left(\mathbb{I}\right)$ and $\{t_{l,m_n}:1\le l\le m_n\}$ be a monotone non-decreasing sequence of non-negative numbers (in sign $t_{l,m_n}\uparrow$). Let the sequence $\{m_n:n\in \mathbb{P}\}$ be such that the next condition holds

$$\underset{n}{sup}V(m_n,\left\{t_{l,m_n}\right\})<\infty .$$

Then, the subsequence $t_{m_n}\left(f\right)$ converges in the norm of the space $X\left(\mathbb{I}\right)$.

The next theorem proofs the sharpness of condition (41).

Theorem 4.

Let the sequences $\{t_{l,n}:1\le l\le n\}$ be monotone non-decreasing (in sign $t_{l,n}\uparrow$) for all $n\in \mathbb{P}$. Let $\{m_A:A\in \mathbb{N}\}$ be a sequence of natural numbers such that

$$\underset{A}{sup}V(m_A,\left\{t_{l,m_A}\right\})=\infty .$$

Then, there exists a sequence $\{p_j:j\in \mathbb{N}\}$ and a function $f\in X\left(\mathbb{I}\right)$ such that

$$\omega {\left(\frac{1}{2^{|m_{p_j}|}},f\right)}_X=O\left(\frac{1}{V(m_{p_j},\left\{t_{l,m_{p_j}}\right\})}\right)$$

and

$$\parallel t_{m_{p_l}}{\left(f\right)-f\parallel }_X\nrightarrow 0\mathrm{as}l\to \infty .$$

Proof of Theorem 4.

Let the sequence $\{t_{k,n}:1\le k\le n\}$ be monotone non-decreasing (in sign $t_{k,n}\uparrow$) for all $n\in \mathbb{P}$. Then, condition

$$\underset{n}{sup}V\left(m_n,\left\{t_{k,m_n}\right\}\right)=\infty$$

yields that there exists a sequence $\left\{p_l:l\in \mathbb{N}\right\}$ such that the following two conditions hold

$$\left|m_{p_l}\right|>\left|m_{p_{l-1}}\right|+2log\left(l+1\right)$$

(42)

and

$$V\left(m_{p_l},\left\{t_{s,m_{p_l}}\right\}\right)\ge 2lV\left(m_{p_{l-1}},\left\{t_{s,m_{p_{l-1}}}\right\}\right).$$

(43)

First, let us discuss $X\left(\mathbb{I}\right)=L_1\left(\mathbb{I}\right)$. Now, we set

$$g\left(x\right):=\sum _{j=1}^{\infty }{g}_j\left(x\right),g_j\left(x\right):=\frac{D_{2^{|m_{p_j}|+1}}\left(x\right)}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}.$$

It is easy to check that $g\in L_1\left(\mathbb{I}\right)$. Let us calculate $\omega {\left(\frac{1}{m_{p_k}},g\right)}_{L_1}$. We set $y\in I_{|m_{p_k}|}$, and we learn that

$$D_{2^{|m_{p_l}|}}\left(x\dotplus y\right)-D_{2^{|m_{p_l}|}}\left(x\right)=0\mathrm{for}l=1,2,\dots ,k-1.$$

(44)

Inequalities (43) and (44) yield that

$$\begin{array}{ccc}& & \underset{\mathbb{I}}{\int }\left|g\left(x\dotplus y\right)-g\left(x\right)\right|dx\hfill \\ & \le & \sum _{j=k}^{\infty }\frac{1}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}\underset{\mathbb{I}}{\int }\left|D_{2^{|m_{p_j}|+1}}\left(x\dotplus y\right)-D_{2^{|m_{p_j}|+1}}\left(x\right)\right|dx\hfill \\ & \le & \frac{c}{V\left(m_{p_k},\left\{t_{s,m_{p_k}}\right\}\right)}.\hfill \end{array}$$

Consequently, taking the supremum for all $y\in I_{|m_{p_k}|}$, we have that

$$\omega {\left(\frac{1}{m_{p_k}},g\right)}_{L_1}=O\left(\frac{1}{V\left(m_{p_k},\left\{t_{s,m_{p_k}}\right\}\right)}\right).$$

We can write

$$\begin{array}{ccc}\hfill {∥t_{m_{p_k}}\left(g\right)-g∥}_1& \ge & {∥t_{m_{p_k}}\left(\sum _{j=k}^{\infty }{g}_j\right)∥}_1-\sum _{j=k}^{\infty }{∥g_j∥}_1\hfill \\ & & -{∥t_{m_{p_k}}\left(\sum _{j=1}^{k-1}{g}_j\right)-\sum _{j=1}^{k-1}{g}_j∥}_1.\hfill \end{array}$$

(45)

For $j\ge k$

$$\begin{array}{ccc}\hfill t_{m_{p_k}}\left(g_j\right)& =& g_j\ast F_{m_{p_k}}=F_{m_{p_k}}\ast g_j=\frac{1}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}{S}_{2^{|m_{p_j}|+1}}\left(F_{m_{p_k}}\right)\hfill \\ & =& \frac{1}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}{F}_{m_{p_k}}.\hfill \end{array}$$

From inequality (19), we have that

$${∥t_{m_{p_k}}\left(\sum _{j=k}^{\infty }{g}_j\right)∥}_1=\sum _{j=k}^{\infty }\frac{{∥F_{m_{p_k}}∥}_1}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}\ge \frac{{∥F_{m_{p_k}}∥}_1}{V\left(m_{p_k},\left\{t_{s,m_{p_k}}\right\}\right)}\ge 1>0.$$

(46)

Equality (3) and condition (43) yield that

$$\sum _{j=k}^{\infty }{∥g_j∥}_1\le \sum _{j=k}^{\infty }\frac{1}{V\left(m_{p_j},\left\{t_{s,m_{p_j}}\right\}\right)}\le \frac{2}{V\left(m_{p_k},\left\{t_{s,m_{p_k}}\right\}\right)}.$$

(47)

By Theorem 3 and (44), we obtain the following inequality $\left(j<k\right)$

$$\begin{array}{ccc}\hfill {∥t_{m_{p_k}}\left(g_j\right)-g_j∥}_1& \le & c\sum _{r=0}^{|m_{p_k}|-2}{2}^r{t}_{2^{r+1}-1,m_{p_k}}\omega {\left(\frac{1}{2^r},g_j\right)}_{L_1}\hfill \\ & \le & c\sum _{r=0}^{|m_{p_j}|-1}{2}^r{t}_{2^{r+1}-1,m_{p_k}}\omega {\left(\frac{1}{2^r},g_j\right)}_{L_1}\hfill \\ & \le & c\sum _{r=0}^{|m_{p_j}|-1}{2}^r{t}_{2^{r+1}-1,m_{p_k}}\hfill \\ & \le & c{t}_{2^{|m_{p_k}|-1},m_{p_k}}\sum _{r=0}^{|m_{p_j}|-1}{2}^r.\hfill \end{array}$$

Since the sequence $\left\{t_{s,m_{p_k}}\right\}$ is non-decreasing, we write

$$2^{|m_{p_k}|-1}{t}_{2^{|m_{p_k}|-1},m_{p_k}}\le \sum _{s=2^{|m_{p_k}|-1}}^{2^{|m_{p_k}|}-1}{t}_{s,m_{p_k}}\le \sum _{s=1}^{m_{p_k}}{t}_{s,m_{p_k}}=1$$

and

$$t_{2^{|m_{p_k}|-1},m_{p_k}}\le \frac{1}{2^{|m_{p_k}|-1}}.$$

(48)

By inequality (42), we obtain

$${∥t_{m_{p_k}}\left(g_j\right)-g_j∥}_1\le \frac{c}{2^{|m_{p_k}|-1}}\sum _{r=0}^{|m_{p_j}|-1}{2}^r\le \frac{c}{k^2}$$

and

$$\sum _{j=1}^{k-1}{∥t_{m_{p_k}}\left(g_j\right)-g_j∥}_1\le \frac{c}{k}.$$

(49)

Combining (45)–(49), we have that

$$\overline{\underset{k\to \infty }{lim}}{∥t_{m_{p_k}}\left(g\right)-g∥}_1>0.$$

Second, we discuss the case $X\left(\mathbb{I}\right)=C_W\left(\mathbb{I}\right)$. Let the condition (42) and (43) hold as well. We define the function h by

$$h\left(x\right):=\sum _{j=1}^{\infty }\frac{h_j\left(x\right)}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)},$$

where

$$h_j\left(x\right):=\mathrm{sgn}\left(F_{m_{p_j}}\right).$$

It is easily seen that $h\in C_w\left(\mathbb{I}\right)$. Now, we calculate the modulus of continuity in $C_W$. Let $y\in I_{|m_{p_k}|},$ then for $j=1,2,\dots ,k-1$, we obtain

$$h_j\left(x\dotplus y\right)-h_j\left(x\right)=0.$$

Applying condition (43), we obtain

$$\begin{array}{ccc}\hfill \left|h\left(x\dotplus y\right)-h\left(x\right)\right|& \le & 2\sum _{j=k}^{\infty }\frac{1}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)}\hfill \\ & =& O\left(\frac{1}{V\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)}\right).\hfill \end{array}$$

That is,

$$\omega {\left(\frac{1}{m_{p_k}},h\right)}_{C_W}=O\left(\frac{1}{V\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)}\right).$$

It is easily seen that

$$\begin{array}{ccc}\hfill \left|t_{m_{p_k}}\left(h,0\right)-h\left(0\right)\right|& \ge & \frac{\left|t_{m_{p_k}}\left(h_k,0\right)\right|}{V\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)}-\sum _{j=k}^{\infty }\frac{\left|h_j\left(0\right)\right|}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)}\hfill \\ & -& \sum _{j=k+1}^{\infty }\frac{\left|t_{m_{p_k}}\left(h_j,0\right)\right|}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)}-\sum _{j=1}^{k-1}\frac{\left|t_{m_{p_k}}\left(h_j,0\right)-h_j\left(0\right)\right|}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)}\hfill \\ \hfill =:& & Q_1-Q_2-Q_3-Q_4.\hfill \end{array}$$

(50)

Theorem 1, conditions (42) and (43) yield that

$$Q_1\ge \frac{{∥F_{m_{p_k}}∥}_1}{V\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)}\ge c>0,$$

(51)

$$Q_2\le \frac{c}{V\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)},$$

(52)

$$Q_3\le \sum _{j=k+1}^{\infty }\frac{{∥F_{m_{p_k}}∥}_1}{V\left(m_{p_j},\left\{t_{l,m_{p_j}}\right\}\right)}\le \frac{cV\left(m_{p_k},\left\{t_{l,m_{p_k}}\right\}\right)}{V\left(m_{p_{k+1}},\left\{t_{l,m_{p_{k+1}}}\right\}\right)}\le \frac{c}{k}.$$

(53)

We apply Theorem 3, inequality (48), conditions (42) and (43); we have that

$$\begin{array}{ccc}\hfill Q_4& \le & c\sum _{j=1}^{k-1}\sum _{r=0}^{|m_{p_k}|-2}{2}^r{t}_{2^{r+1}-1,m_{p_k}}\omega {\left(\frac{1}{2^r},h_j\right)}_{C_W}\hfill \\ & \le & c\sum _{j=1}^{k-1}\sum _{r=0}^{|m_{p_j}|-1}{2}^r{t}_{2^{r+1}-1,m_{p_k}}\omega {\left(\frac{1}{2^r},h_j\right)}_{C_W}\hfill \\ & \le & c\sum _{j=1}^{k-1}{t}_{2^{|m_{p_k}|-1},m_{p_k}}\sum _{r=0}^{|m_{p_j}|-1}{2}^r\hfill \\ & \le & c\sum _{j=1}^{k-1}\frac{1}{2^{|m_{p_k}|-1}}\sum _{r=0}^{|m_{p_j}|-1}{2}^r\hfill \\ & \le & \frac{ck{2}^{|m_{p_{k-1}}|}}{2^{|m_{p_k}|}}\le \frac{c}{k}.\hfill \end{array}$$

(54)

Combining (50)–(54), we complete the proof of Theorem 4. □

8. Almost Everywhere Convergence of Matrix Transforms of Walsh–Fourier Series

Let us set $E_n\left(f;x\right)=S_{2^n}\left(f;x\right)$. The maximal function is defined by

$$E^{\ast }\left(f;x\right)=\underset{n\in \mathbb{N}}{sup}\left|E_n\left(f;x\right)\right|.$$

It is known that ([1], p. 81) there exists a positive constant c such that

$$y\left|\{x\in \mathbb{I}:E^{*}\left(f;x\right)>y\}\right|\le c{\parallel f\parallel }_1$$

(55)

holds for all $f\in L_1\left(\mathbb{I}\right)$ and $y>0$.

We define the maximal operator $t^{*}$ of the linear transforms $t_n$ generated by the sequences $\{t_{k,n}:1\le k\le n\}$

$$t^{*}\left(f\right):=\underset{n}{sup}\left|t_n\left(f\right)\right|.$$

In this section, we discuss some properties of the maximal operator $t^{*}\left(f\right)$. As a consequence, we learn that the matrix transforms $t_n\left(f\right)$ of the Walsh–Fourier series converge almost everywhere to the function f for all integrable functions. This result is reached with different monotonity conditions.

First, we state the boundedness of the maximal operator of the linear transforms defined by monotone non-increasing sequences.

Theorem 5.

Let $\{t_{k,n}:1\le k\le n\}$ be monotone non-increasing sequences of non-negative numbers (in sign $t_{k,n}\downarrow$) for all $n\in \mathbb{P}$. Then, the maximal operator $t^{*}$ is bounded from the Lebesque space $L_p$ to the Lebesque space $L_p$ for all $1<p\le \infty$. That is, there exists a positive constant $C_p$ which depends only on p such that

$$\parallel t^{*}{\left(f\right)\parallel }_p\le C_p{\parallel f\parallel }_p$$

holds for all $f\in L_p\left(\mathbb{I}\right)$. Moreover, the maximal operator $t^{*}$ is of weak type $(1,1)$. That is, there exists a positive constant c such that

$$\underset{\lambda >0}{sup}\lambda \left|\{t^{*}\left(f\right)>\lambda \}\right|:={∥t^{\ast }\left(f\right)∥}_{weak-L_1\left(\mathbb{I}\right)}\le c{\parallel f\parallel }_1$$

holds for all $f\in L_1\left(\mathbb{I}\right)$, $\lambda >0$.

Proof of Theorem 5.

Since (see (20))

$$f\ast F_n=\sum _{k=1}^{n-1}(t_{k,n}-t_{k+1,n})k\left(f\ast K_k\right)+t_{n,n}n\left(f\ast K_n\right),$$

$$\underset{\lambda >0}{sup}\lambda \left|\{\underset{k}{sup}\left|f\ast K_k\right|>\lambda \}\right|\le c{\parallel f\parallel }_1,f\in L_1\left(\mathbb{I}\right)$$

$${∥\underset{k}{sup}\left|f\ast K_k\right|∥}_p\le c{\parallel f\parallel }_p,p\ge 1,f\in L_p\left(\mathbb{I}\right)$$

and

$$\begin{array}{ccc}\hfill \underset{n}{sup}\left|f\ast F_n\right|& \le & c\underset{k}{sup}\left|f\ast K_k\right|\left(\sum _{k=1}^{n-1}(t_{k,n}-t_{k+1,n})k+t_{n,n}n\right)\hfill \\ & \le & c\underset{k}{sup}\left|f\ast K_k\right|,\hfill \end{array}$$

we complete the proof of Theorem 5. □

By the well-known density argument due to Marcinkiewicz and Zygmund [9], the next corollary holds.

Corollary 4.

Let $\{t_{k,n}:1\le k\le n\}$ be a monotone non-increasing sequence of non-negative numbers (in sign $t_{k,n}\downarrow$) for all $n\in \mathbb{P}$ and $f\in L_1\left(\mathbb{I}\right)$. Then

$$\underset{n\to \infty }{lim}{t}_n(f;x)=f\left(x\right)fora.e.x\in \mathbb{I}.$$

Now, we consider the following maximal operator

$$\underset{n}{sup}\left|f\ast \left|K_n\right|\right|.$$

We prove that the maximal operator is of weak (1,1) type. That is, there exists a positive constant c such that

$$\underset{\lambda >0}{sup}\lambda \left|\{\underset{n}{sup}\left|f\ast \left|K_n\right|\right|>\lambda \}\right|\le c{\parallel f\parallel }_1$$

(56)

holds for all $f\in L_1\left(\mathbb{I}\right)$, $\lambda >0$. For this, it is enough to prove that the operator $\underset{n}{sup}\left|f\ast \left|K_n\right|\right|$ is quasi-local and bounded from the space $L_{\infty }\left(I\right)$ to the space $L_{\infty }\left(I\right)$ (see [1]). The boundedness immediately follows from (7). Now, we prove the quasi-locality. In particular, let $f\in L_1\left(\mathbb{I}\right)$ such that $\mathrm{supp}\left(f\right)\subset I_N\left(u^{\prime }\right),\underset{I_N\left(u^{\prime }\right)}{\int }f=0$ for some dyadic interval $I_N\left(u^{\prime }\right)$. Then, we show that there exists a positive constant c such that the next inequality

$$\underset{\overline{I_N\left(u^{\prime }\right)}}{\int }\underset{n}{sup}\left|f\ast \left|K_n\right|\right|\le c{∥f∥}_1$$

holds. It can be supposed that $u^{\prime }=0$. If $n\le 2^N$, then

$$\left|f\ast \left|K_n\right|\right|=0.$$

Consequently, $n>2^N$ can be supposed.

It is known that (see Gát [10])

$$\underset{{\overline{I}}_N}{\int }\underset{n\ge 2^N}{sup}\left|K_n\right|<\infty ,$$

(57)

Then, we have

$$\begin{array}{ccc}& & \underset{{\overline{I}}_N}{\int }\underset{n\ge 2^N}{sup}\left|\left(f\ast \left|K_n\right|\right)\left(x\right)\right|dx\hfill \\ & =& \underset{{\overline{I}}_N}{\int }\underset{n\ge 2^N}{sup}\left|\underset{\mathbb{I}}{\int }f\left(t\right)\left|K_n\left(x\dotplus t\right)\right|dt\right|dx\hfill \\ & \le & \underset{\mathbb{I}}{\int }\left|f\left(t\right)\right|\underset{{\overline{I}}_N}{\int }\underset{n\ge 2^N}{sup}\left|K_n\left(x\dotplus t\right)\right|dxdt\hfill \\ & \le & c{∥f∥}_1.\hfill \end{array}$$

Hence, (56) is proved.

From (24), we can write

$$\begin{array}{ccc}\hfill F_{n,2}& =& w_n\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n_{\left(s\right)}}{D}_{2^s}\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}\hfill \\ & & -w_n\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)w_{n_{\left(s\right)}}{w}_{2^s-1}\hfill \\ & & \times \left(\sum _{l=1}^{2^s-2}(t_{n^{\left(s\right)}-l,n}-t_{n^{\left(s\right)}-l-1,n})l{K}_l+t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)K_{2^s-1}\right).\hfill \end{array}$$

Let us set

$$\begin{array}{ccc}\hfill {\tilde{F}}_{n,2}& :& =\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)D_{2^s}\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-2}(t_{n^{\left(s\right)}-l,n}-t_{n^{\left(s\right)}-l-1,n})l\left|K_l\right|\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)\left|K_{2^s-1}\right|.\hfill \end{array}$$

It is easy to see that

$$\left|F_{n,2}\right|\le {\tilde{F}}_{n,2}.$$

In order to prove Theorem 6, we need the following lemmas.

Lemma 4.

Let $\{t_{k,n}:k=1,\dots ,n\}$ be a monotone non-decreasing sequence of non-negative numbers for every fixed $n\in \mathbb{N}$. The operator $\underset{n\in \mathbb{N}}{sup}\left|f\ast {\tilde{F}}_{n,2}\right|$ is of weak type $\left(1,1\right)$. That is, there exists a positive constant c such that

$$\underset{\lambda >0}{sup}\lambda \left|\{\underset{n\in \mathbb{N}}{sup}\left|f\ast {\tilde{F}}_{n,2}\right|>\lambda \}\right|\le c{\parallel f\parallel }_1$$

holds for all $f\in L_1\left(\mathbb{I}\right)$, $\lambda >0$.

Proof of Lemma 4.

We can write

$$\begin{array}{ccc}& & \left|f\ast {\tilde{F}}_{n,2}\right|\hfill \\ & \le & \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}\left|f\ast D_{2^s}\right|\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-2}(t_{n^{\left(s\right)}-l,n}-t_{n^{\left(s\right)}-l-1,n})l\left|f\ast \left|K_l\right|\right|\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)\left|f\ast \left|K_{2^s-1}\right|\right|\hfill \end{array}$$

(58)

Since

$$\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-1}{t}_{n^{\left(s\right)}-l,n}=\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=n^{\left(s+1\right)}+1}^{n^{\left(s\right)-1}}{t}_{l,n}\le \sum _{k=1}^n{t}_{k,n}\le c<\infty$$

and

$$\begin{array}{ccc}& & \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-2}(t_{n^{\left(s\right)}-l,n}-t_{n^{\left(s\right)}-l-1,n})l\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=1}^{2^s-2}\left[(l{t}_{n^{\left(s\right)}-l,n}-\left(l+1\right)t_{n^{\left(s\right)}-l-1,n})+t_{n^{\left(s\right)}-l-1,n}\right]\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)t_{n^{\left(s\right)}-2^s+1,n}(2^s-1)\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\left[(t_{n^{\left(s\right)}-1,n}-\left(2^s-1\right)t_{n^{\left(s+1\right)}+1,n})+\sum _{l=1}^{2^s-2}{t}_{n^{\left(s\right)}-l-1,n}\right]\hfill \\ & & +\sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)t_{n^{\left(s+1\right)}+1,n}(2^s-1)\hfill \\ & =& \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=0}^{2^s-2}{t}_{n^{\left(s\right)}-l-1,n}\hfill \\ & \le & \sum _{s=0}^{\left|n\right|}{\epsilon }_s\left(n\right)\sum _{l=n^{\left(s\right)}-1}^{n^{\left(s+1\right)}}{t}_{l,n}\le \sum _{k=1}^n{t}_{k,n}\le c<\infty ,\hfill \end{array}$$

from (58), we have

$$\underset{n\in \mathbb{N}}{sup}\left|f\ast {\tilde{F}}_{n,2}\right|\le c\left(\underset{k\in \mathbb{N}}{sup}\left|f\ast \left|K_k\right|\right|+E^{*}\left(f\right)\right)$$

and consequently, by (56) and (55), we complete the proof of Lemma 4. □

Theorem 6.

Let $\{m_A:A\in \mathbb{P}\}$ be a strictly monotone increasing sequence. Let $\{t_{k,n}:k=1,\dots ,n\}$ be a monotone non-decreasing sequence of non-negative numbers for every fixed $n\in \mathbb{N}$. If

$$\underset{A\in \mathbb{P}}{sup}V\left(m_A,\left\{t_{k,m_A}\right\}\right)<\infty$$

(59)

holds, then there exists a positive constant c such that

$$\underset{\lambda >0}{sup}\lambda \left|\{\underset{A}{sup}|t_{m_A}\left(f\right)|>\lambda \}\right|\le c{\parallel f\parallel }_1$$

holds for all $f\in L_1\left(\mathbb{I}\right)$, $\lambda >0$.

Proof of Theorem 6.

We have (see (9))

$$\begin{array}{ccc}\hfill t_n\left(f\right)& =& f\ast w_n{F}_n^{*}\hfill \\ & =& f\ast w_n{F}_{n,1}^{*}+f\ast w_n{F}_{n,2}^{*}.\hfill \end{array}$$

(60)

We obtain

$$\begin{array}{ccc}& & \left|f\ast w_n{F}_{n,1}^{*}\right|\hfill \\ & =& \left|\sum _{k=0}^{\infty }{\alpha }_k\left(n\right)(S_{2^{k+1}}\left(f{w}_n\right)-S_{2^k}\left(f{w}_n\right))\right|\hfill \\ & \le & \left|\sum _{k=0}^{\infty }\left({\alpha }_k\left(n\right)-{\alpha }_{k+1}\left(n\right)\right)S_{2^{k+1}}\left(f{w}_n\right)\right|+\left|{\alpha }_0\left(n\right)S_{2^0}\left(f{w}_n\right)\right|\hfill \\ & \le & E^{\ast }\left(\left|f\right|\right)\sum _{k=0}^{\infty }\left|{\alpha }_k\left(n\right)-{\alpha }_{k+1}\left(n\right)\right|+\left|{\alpha }_0\left(n\right)\right|\hfill \end{array}$$

where ${\alpha }_k\left(n\right)={\epsilon }_k\left(n\right)T_n^{\left(n^{\left(k\right)}\right)}$. Since

$$\begin{array}{ccc}& & \underset{n\in \mathbb{N}}{sup}\left(\sum _{k=0}^{\infty }\left|{\alpha }_k\left(n\right)-{\alpha }_{k+1}\left(n\right)\right|+\left|{\alpha }_0\left(n\right)\right|\right)\hfill \\ & \le & \underset{n\in \mathbb{N}}{sup}\left(\sum _{k=0}^{\infty }\left|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)\right|T_n^{\left(n^{\left(k\right)}\right)}\right)+c<\infty ,\hfill \end{array}$$

we conclude that

$$\underset{n\in \mathbb{N}}{sup}\left|f\ast w_n{F}_{n,1}^{*}\right|\le c{E}^{*}\left(\left|f\right|,x\right).$$

Consequently, we can write

$${∥\underset{n\in \mathbb{N}}{sup}\left|f\ast w_n{F}_{n,1}^{*}\right|∥}_{\mathrm{weak}-L_1\left(\mathbb{I}\right)}\le c{∥f∥}_1.$$

(61)

By Lemma 4, we obtain

$${∥\underset{n\in \mathbb{N}}{sup}\left|f\ast w_n{F}_{n,2}^{*}\right|∥}_{\mathrm{weak}-L_1\left(\mathbb{I}\right)}\le c{∥f∥}_1.$$

(62)

We combine (60), (61) and (62) in order to obtain

$${∥\underset{n\in \mathbb{N}}{sup}\left|t_n\left(f\right)\right|∥}_{\mathrm{weak}-L_1\left(\mathbb{I}\right)}\le c{∥f∥}_1.$$

Theorem 6 is proved. □

Let us define for positive real numbers K the subset $L_K\left(\left\{t_{k,n}\right\}\right)$ of natural numbers by

$$L_K\left(\left\{t_{k,n}\right\}\right):=\left\{n\in \mathbb{N}:\sum _{k=1}^{\left|n\right|}|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)|T^{\left(n^{\left(k\right)}\right)}\le K\right\}.$$

The next corollary follows from Theorem 6 by the well-known density argument due to Marcinkiewicz and Zygmund [9].

Corollary 5.

Let $\{t_{k,n}:k=1,\dots ,n\}$ be a monotone non-decreasing sequence of non-negative numbers for every fixed $n\in \mathbb{N}$ and $f\in L_1\left(\mathbb{I}\right)$. Then, $t_n(f;x)\to f$ almost everywhere provided that $n\to \infty$ and $n\in L_K\left(\left\{t_{k,n}\right\}\right)$.

9. Application: Cesàro Means with Varying Parameters of Walsh–Fourier Series

The theorems can be used for various methods of summability. In this section, the application of the theorems proved above to Cesàro means with varying parameters will be presented.

The $\left(C,{\alpha }_n\right)$ means of the Walsh–Fourier series of the function f is given by

$${\sigma }_n^{{\alpha }_n}(f,x)=\frac{1}{A_{n-1}^{{\alpha }_n}}\sum _{j=1}^n{A}_{n-j}^{{\alpha }_n-1}{S}_j(f,x),$$

where

$$A_n^{{\alpha }_n}:=\frac{(1+{\alpha }_n)\dots (n+{\alpha }_n)}{n!}$$

for any $n\in \mathbb{N},{\alpha }_n\ne -1,-2,\dots$. The $\left(C,{\alpha }_n\right)$ kernel is defined by

$$K_n^{{\alpha }_n}=\frac{1}{A_{n-1}^{{\alpha }_n}}\sum _{j=1}^n{A}_{n-j}^{{\alpha }_n-1}{D}_j.$$

We shall need the following Lemma (see [11]).

Lemma 5.

Let $k,n\in \mathbb{N}$. Then

$$c_1\left(d\right)k^{{\alpha }_n}<A_k^{{\alpha }_n}<c_2\left(d\right)k^{{\alpha }_n},0<{\alpha }_n\le d.$$

(63)

The idea of Cesàro means with variable parameters of numerical sequences is due to Kaplan [12], and the introduction of these $\left(C,{\alpha }_n\right)$ means of Fourier series is due to Akhobadze [11].

The almost everywhere convergence of the subsequence of Cesàro means with variable parameters has been studied by the following authors: Abu Joudeh and Gát [6], Gát and Goginava [13,14], Weisz [15].

Let $t_{k,n}=A_{n-k}^{{\alpha }_n-1}/A_{n-1}^{{\alpha }_n},0\le k\le n.$ Then, from (63), we have

$$T_n^{\left(n^{\left(s\right)}\right)}=\sum _{l=n^{\left(s\right)}}^n\frac{A_{n-l}^{{\alpha }_n-1}}{A_{n-1}^{{\alpha }_n}}=\sum _{l=0}^{n_{\left(s-1\right)}}\frac{A_l^{{\alpha }_n-1}}{A_{n-1}^{{\alpha }_n}}=\frac{A_{n_{\left(s-1\right)}}^{{\alpha }_n}}{A_{n-1}^{{\alpha }_n}}\le \frac{c{2}^{s{\alpha }_n}}{2^{\left|n\right|{\alpha }_n}}.$$

Hence, from Corollary 5, we obtain

Theorem 7

(see [14]). Suppose that ${\alpha }_n\in \left(0,1\right)$. Let $f\in L_1\left(\mathbb{I}\right)$. Then, ${\sigma }_n^{{\alpha }_n}\left(f\right)\to f$ almost everywhere provided that $n\to \infty$ and $n\in L_K\left(\left\{A_{n-k}^{{\alpha }_n-1}/A_{n-1}^{{\alpha }_n}\right\}\right)$.

Now, we consider the rate of convergence of the Cesàro means with varying parameters of Walsh–Fourier series. Since

$$V(n,\left\{t_{k,n}\right\})\le \frac{c}{2^{\left|n\right|{\alpha }_n}}\sum _{k=1}^{\left|n\right|}|{\epsilon }_k\left(n\right)-{\epsilon }_{k+1}\left(n\right)|2^{k{\alpha }_n}\le \frac{c}{{\alpha }_n}$$

and (see Lemma 5)

$$t_{2^{r+1}-1,n}=\frac{A_{n-2^{r+1}+1}^{{\alpha }_n-1}}{A_n^{{\alpha }_n}}\sim {\alpha }_n\frac{{\left(n-2^{r+1}+1\right)}^{{\alpha }_n-1}}{n^{{\alpha }_n}}\sim {\alpha }_n{2}^{-\left|n\right|}$$

from Theorem 3, we have

Theorem 8.

Let $f\in X\left(\mathbb{I}\right)$ and ${\alpha }_n\in \left(0,1\right)$. Then,

$$\begin{array}{ccc}\hfill \parallel {\sigma }_n^{{\alpha }_n}\left(f\right){-f\parallel }_X& \le & \frac{c_1}{{\alpha }_n}\omega {\left(\frac{1}{2^{\left|n\right|}},f\right)}_X+c_2\omega {\left(\frac{1}{2^{\left|n\right|-1}},f\right)}_X\hfill \\ & & +c_3{\alpha }_n\sum _{r=0}^{\left|n\right|-2}{2}^{r-\left|n\right|}\omega {\left(\frac{1}{2^r},f\right)}_X.\hfill \end{array}$$