Cesàro means with varying parameters of Walsh–Fourier series

Abstract

In this paper we discuss some convergence and divergence properties of subsequences of Cesàro means with varying parameters of Walsh–Fourier series. We give necessary and sufficient conditions for the convergence regarding the weighted variation of numbers.

1 Walsh functions

We shall denote the set of all nonnegative integers by $\mathbb {N}$, the set of all integers by $\mathbb {Z}$ and the set of dyadic rational numbers in the unit interval $\mathbb {I}:=[0,1)$ by $\mathbb {Q}$. In particular, each element of $\mathbb {Q}$ has the form $\frac{p}{2^{n}}$ for some $p,n\in \mathbb {N}$, $0\le p\le 2^{n}$.

Denote the dyadic enpansion of $n\in \mathbb {N}$ and $x\in \mathbb {I}$ by

$$\begin{aligned} n=\sum \limits _{j=0}^{\infty }\varepsilon _{j}\left( n\right) 2^{j},\varepsilon _{j}\left( n\right) =0,1, \end{aligned}$$

and

$$\begin{aligned} x=\sum \limits _{j=0}^{\infty }\frac{x_{j}}{2^{j+1}},x_{j}=0,1. \end{aligned}$$

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

$$\begin{aligned}{} & {} x\dotplus y:=\sum \limits _{n=0}^{\infty }\left| x_{n}-y_{n}\right| 2^{-\left( n+1\right) },\nonumber \\{} & {} k\oplus n:=\sum \limits _{i=0}^{\infty }\left| \varepsilon _{i}\left( k\right) -\varepsilon _{i}\left( n\right) \right| 2^{i}, \end{aligned}$$

(1.1)

and by the definition of $w_{n}$ we have

$$\begin{aligned} w_{k\oplus n}=w_{k}w_{n}. \end{aligned}$$

(1.2)

The sets $I_{n}\left( x\right) :=\left\{ y\in \mathbb {I}:y_{0}=x_{0},\ldots ,y_{n-1}=x_{n-1}\right\}$ for $x\in \mathbb {I},I_{n}:=I_{n}\left( 0\right)$ for $0<n\in \mathbb {N}$ and $I_{0}\left( x\right) :=\mathbb {I}$ are the dyadic intervals of $\mathbb {I}$. For $0<n\in \mathbb {N}$ let $\left| n\right| :=\max \left\{ j\in \mathbb {N}:n_{j}\ne 0\right\}$, that is, $2^{\left| n\right| }\le n<2^{\left| n\right| +1}.$

The Rademacher system is defined by

$$\begin{aligned} \rho _{n}\left( x\right) :=\left( -1\right) ^{x_{n}} \ \ \ \left( x\in \mathbb {I},n\in \mathbb {N}\right) . \end{aligned}$$

The Walsh–Paley system is defined as the sequence of the Walsh–Paley functions:

$$\begin{aligned} w_{n}\left( x\right) :=\prod \limits _{k=0}^{\infty }\left( \rho _{k}\left( x\right) \right) ^{\varepsilon _{k}\left( n\right) }=\left( -1\right) ^{\sum \limits _{k=0}^{\left| n\right| }\varepsilon _{k}\left( n\right) x_{k}}\left( x\in \mathbb {I},n\in \mathbb {N}\right) . \end{aligned}$$

The Walsh–Dirichlet kernel is defined by

$$\begin{aligned} D_{n}\left( x\right) := \sum \limits _{k=0}^{n-1}w_{k}\left( x\right) \left( n\in \mathbb {N}\right) , D_{0}=0. \end{aligned}$$

Recall that (see [23])

$$\begin{aligned} D_{2^{n}}\left( x\right) =\left\{ \begin{array}{ll} 2^{n}, &{} \text{ if } x\in I_{n}\left( 0\right) , \\ 0, &{} \text{ if } x\in \mathbb {I}\backslash I_{n}\left( 0\right) . \end{array} \right. \end{aligned}$$

(1.3)

As usual, denote by $L_{1}(\mathbb {I})$ the set of measurable functions defined on $\mathbb {I}$ for which

$$\begin{aligned} \left\| f\right\| _{1}:=\int \limits _{\mathbb {I}}\left| f\left( t\right) \right| \,dt <\infty \text {.} \end{aligned}$$

Let $f\in L_{1}(\mathbb {I})$. The partial sums of the Walsh–Fourier series are defined as follows:

$$\begin{aligned} S_{M}\left( x,f\right) :=\sum \limits _{i=0}^{M-1}\widehat{f}\left( i\right) w_{i}\left( x\right) , \end{aligned}$$

where the number

$$\begin{aligned} \widehat{f}\left( i\right) =\int \limits _{\mathbb {I}}f\left( t\right) w_{i}\left( t\right) \,dt \end{aligned}$$

is said to be the ith Walsh–Fourier coefficient of the function f. Set

$$\begin{aligned} E_{n}\left( x, f\right) := S_{2^{n}}\left( x,f\right) . \end{aligned}$$

The maximal function is defined by

$$\begin{aligned} E^{*}\left( x,f\right) := \sup \limits _{n\in \mathbb {N}}E_{n}\left( x,\left| f\right| \right) . \end{aligned}$$

2 Cesàro means with varying parameters

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

$$\begin{aligned} \sigma _{n}^{\alpha _{n}}(f,x) := \frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=1}^{n}A_{n-j}^{\alpha _{n}-1}S_{j}(f,x)=\frac{1}{ A_{n-1}^{\alpha _{n}}}\sum \limits _{j=0}^{n-1}A_{n-j}^{\alpha _{n}}\widehat{f} \left( j\right) w_{j}\left( x\right) , \end{aligned}$$

where

$$\begin{aligned} A_{n}^{\alpha _{n}}:=\frac{(1+\alpha _{n})\cdots (n+\alpha _{n})}{n!} \end{aligned}$$

for any $n\in \textbf{N},\alpha _n \ne -1,-2,\ldots$. It is known that [29]

$$\begin{aligned} A_{n}^{\alpha _{n}}=\sum \limits _{k=0}^{n}A_{k}^{\alpha _{n}-1},A_{n}^{\alpha _{n}-1}=\frac{\alpha _{n}}{\alpha _{n}+n}A_{n}^{\alpha _{n}}. \end{aligned}$$

(2.1)

The $\left( C,\alpha _{n}\right)$ kernel is defined by

$$\begin{aligned} K_{n}^{\alpha _{n}}=\frac{1}{A_{n-1}^{\alpha }}\sum \limits _{j=1}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}=\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=0}^{n-1}A_{n-j-1}^{\alpha _{n}}w_{j}. \end{aligned}$$

The idea of Cesàro means with variable parameters of numerical sequences is due to Kaplan [17] and the introduction of these $\left( C,\alpha _{n}\right)$ means of Fourier series is due to Akhobadze [2, 3] who investigated the behavior of the $L_{1}$-norm convergence of $\sigma _{n}^{\alpha _{n}}\left( f\right) \rightarrow f$ for the trigonometric system.

The a.e. divergence of Cesàro means with varying parameters of Walsh–Fourier series was investigated by Tetunashvili [25]. Abu Joudeh and Gát in [1] proved the almost everywhere convergence (with some restrictions) of the Cesàro $\left( C,\alpha _{n}\right)$ means of integrable functions.

The first result with respect to the a.e. convergence of the Walsh–Fejér means $\sigma _{n}^{\alpha _{n}}f$ for all integrable function f with constant sequence $\alpha _{n}=\alpha >0$ is due to Fine [6] (see also Weisz [27]). On the rate of convergence of Cesàro means in this constant case see the papers of Yano [28] and Fridli [8]. Approximation properties of Cesàro means of negative order with constant sequence was investigated by the second author in [14]. That is why we investigate only the case when $\alpha _n\in (0,1)$ for every n. It is true that in order for the definition of Cesàro mean to make sense we should only suppose that the paremeter is not a negative integer, but for negative parameters there are available divergence results (see, e.g., [14]) and for parameters at least 1 we have the very well-known almost everywhere and norm convergence results of these means. In other words, the only interesting situation is when $\alpha _n\in (0,1)$ for every n.

It is easy to see that

$$\begin{aligned} \sigma _{n}^{\alpha _{n}}(f,x)=\left( f*K_{n}^{\alpha _{n}}\right) \left( x\right) =\int \limits _{\mathbb {I}}f\left( t\right) K_{n}^{\alpha _{n}}\left( x+t\right) \,dt . \end{aligned}$$

The Fejér kernel is defined by

$$\begin{aligned} K_{n}(t) :=\frac{1}{n}\sum \limits _{k=1}^{n}D_{k}(t). \end{aligned}$$

The following estimate was proved by Akhobadze [2, 3]. If $k,n\in \mathbb {N}$, then

$$\begin{aligned}{} & {} c_{1}\left( 1+\alpha _{n}\right) \left( 2+\alpha _{n}\right) k^{\alpha _{n}}< A_{k}^{\alpha _{n}}<c_{2}\left( 1+\alpha _{n}\right) \left( 2+\alpha _{n}\right) k^{\alpha _{n}}, \nonumber \\{} & {} \qquad \text { when} -2< \alpha _{n}<-1; \end{aligned}$$

(2.2)

$$\begin{aligned}{} & {} c_{1}\left( 1+\alpha _{n}\right) k^{\alpha _{n}}<A_{k}^{\alpha _{n}}<c_{2}\left( 1+\alpha _{n}\right) k^{\alpha _{n}},\text { when } -1<\alpha _{n}<0; \end{aligned}$$

(2.3)

$$\begin{aligned}{} & {} c_{1}\left( d\right) k^{\alpha _{n}}<A_{k}^{\alpha _{n}}<c_{2}\left( d\right) k^{\alpha _{n}},\text { when }0<\alpha _{n}\le d\text {.} \end{aligned}$$

(2.4)

3 $L_{1}$-estimation for the kernel $K_{n}^{\alpha _{n}}$

For $n=\sum \limits _{j=0}^{\infty }\varepsilon _{j}\left( n\right) 2^{j},\varepsilon _{j}\left( n\right) =0,1$, we define

$$\begin{aligned} n^{\left( s\right) }:=\sum \limits _{j=s}^{\infty }\varepsilon _{j}\left( n\right) 2^{j},\ n_{\left( s\right) } := n-n^{\left( s+1\right) }=\sum \limits _{j=0}^{s}\varepsilon _{j}\left( n\right) 2^{j}, \ n_{(-1)}:=0. \end{aligned}$$

Theorem 3.1

If $\alpha _{n}\in (0,1)$ and $n\in \mathbb {N}$, then there exist positive constants $c_{1}$ and $c_{2}$ (independent of n) such that

$$\begin{aligned} \frac{c_{1}}{n^{\alpha _{n}}}\sum \limits _{k=0}^{\left| n\right| }\left| \varepsilon _{k}\left( n\right) -\varepsilon _{k+1}\left( n\right) \right| 2^{k\alpha _{n}}\le \left\| K_{n}^{\alpha _{n}}\right\| _{1}\le \frac{c_{2}}{n^{\alpha _{n}}}\sum \limits _{k=0}^{ \left| n\right| }\left| \varepsilon _{k}\left( n\right) -\varepsilon _{k+1}\left( n\right) \right| 2^{k\alpha _{n}}. \end{aligned}$$

Proof of Theorem 3.1

We can write

$$\begin{aligned} A_{n-1}^{\alpha _{n}}K_{n}^{\alpha _{n}}&= \sum \limits _{j=0}^{n-1}A_{n-j-1}^{\alpha _{n}}w_{j}=\sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) \sum \limits _{j=n^{\left( s+1\right) }}^{n^{\left( s\right) }-1}A_{n-j-1}^{\alpha _{n}}w_{j} \nonumber \\&= \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) \sum \limits _{j=0}^{2^{s}-1}A_{n_{\left( s\right) }-j-1}^{\alpha _{n}}w_{j+n^{\left( s+1\right) }} \nonumber \\&= \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s+1\right) }}\sum \limits _{j=0}^{2^{s}-1}A_{n_{\left( s\right) }-j-1}^{\alpha _{n}}w_{j}. \end{aligned}$$

(3.1)

Since in the case of $\varepsilon _s(n)=1$

$$\begin{aligned} n_{\left( s\right) }-j-1=n_{\left( s-1\right) }+2^{s}-1-j, \end{aligned}$$

and

$$\begin{aligned} 2^{s}-1-j=\left( 2^{s}-1\right) \oplus j, \end{aligned}$$

from (1.2) and (3.1) we obtain

$$\begin{aligned} A_{n-1}^{\alpha _{n}}K_{n}^{\alpha _{n}}&= \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s+1\right) }}\sum \limits _{j=0}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}}w_{\left( 2^{s}-1\right) \oplus j} \nonumber \\&= \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s\right) }-1}\sum \limits _{j=0}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}}w_{j}. \end{aligned}$$

(3.2)

Applying Abel’s transformation twice we get

$$\begin{aligned}&\sum \limits _{j=0}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}}w_{j} = \sum \limits _{j=1}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}-2}jK_{j}-A_{n_{\left( s\right) }-1}^{\alpha _{n}-1}2^{s}K_{2^{s}}+A_{n_{\left( s\right) }-1}^{\alpha _{n}}D_{2^{s}}. \end{aligned}$$

(3.3)

Combining (3.2)–(3.3) we obtain

$$\begin{aligned} K_{n}^{\alpha _{n}} =&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s\right) }-1}\sum \limits _{j=1}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}-2}jK_{j}\nonumber \\&- \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s\right) }-1}A_{n_{\left( s\right) }-1}^{\alpha _{n}-1}2^{s}K_{2^{s}} \nonumber \\&+ \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n^{\left( s\right) }-1}A_{n_{\left( s\right) }-1}^{\alpha _{n}}D_{2^{s}} \nonumber \\ =:&Q_{1}(n) +Q_{2}(n) +Q_{3}(n). \end{aligned}$$

(3.4)

From the estimates (2.2)–(2.4) we get

$$\begin{aligned}&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{\left| n\right| }\varepsilon _{s}\left( n\right) \sum \limits _{j=1}^{2^{s}-1}j\left| A_{n_{\left( s-1\right) }+j}^{\alpha _{n}-2}\right| \\&\quad \le \frac{c\alpha _{n}}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{\left| n\right| }\varepsilon _{s}\left( n\right) \sum \limits _{j=1}^{2^{s}-1}j\left( n_{\left( s-1\right) }+j\right) ^{\alpha _{n}-2} \\&\quad \le \frac{c\alpha _{n}}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{\left| n\right| }\varepsilon _{s}\left( n\right) \sum \limits _{j=1}^{2^{s}-1}\left( n_{\left( s-1\right) }+j\right) ^{\alpha _{n}-1} \\&\quad \le \frac{c}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{\left| n\right| }\varepsilon _{s}\left( n\right) \sum \limits _{j=1}^{2^{s}-1}A_{n_{\left( s-1\right) }+j}^{\alpha _{n}-1} \\&\quad = \frac{c}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{\left| n\right| }\varepsilon _{s}\left( n\right) \left( A_{n_{\left( s\right) }}^{\alpha _{n}}-A_{n_{\left( s-1\right) }}^{\alpha _{n}}\right) \\&\quad \le c<\infty . \end{aligned}$$

Since $\sup \limits _{n}\left\| K_{n}\right\| _1 <2$ (see [24]; it even holds $\sup \limits _{n}\left\| K_{n}\right\| _1 = 17/15$, see [26]) from (3.4) we infer

$$\begin{aligned} \left\| Q_{l}(n) \right\| _{1}\le c<\infty ,l=1,2. \end{aligned}$$

(3.5)

Next, we find an upper estimate for $\left\| Q_{3}(n) \right\| _{1}$. We have

$$\begin{aligned} w_{n}Q_{3}(n) =\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) w_{n}w_{n^{\left( s\right) }-1}A_{n_{\left( s\right) }-1}^{\alpha _{n}}D_{2^{s}}. \end{aligned}$$

From (1.2) (we can suppose $\varepsilon _s(n)=1$ otherwise nothing is to be added), we get

$$\begin{aligned} w_{n}w_{n^{\left( s\right) }-1}&= w_{n}w_{n^{\left( s+1\right) }+2^{s}-1}=w_{n}w_{n^{\left( s+1\right) }}w_{2^{s}-1} \\&= w_{n\oplus n^{\left( s+1\right) }}w_{2^{s}-1}=w_{n_{\left( s\right) }}w_{2^{s}-1} \\&= w_{2^{s}}w_{n_{\left( s-1\right) }}w_{2^{s}-1}=w_{2^{s}}w_{n_{\left( s-1\right) \oplus \left( 2^{s}-1\right) }}. \end{aligned}$$

Since $n_{\left( s-1\right) \oplus \left( 2^{s}-1\right) }<2^{s}$, from (1.3) we obtain

$$\begin{aligned} D_{2^{s}}w_{n_{\left( s-1\right) \oplus \left( 2^{s}-1\right) }}=D_{2^{s}}. \end{aligned}$$

Consequently,

$$\begin{aligned} w_{n}Q_{3}(n) =&\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) A_{n_{\left( s\right) }-1}^{\alpha _{n}}w_{2^{s}}D_{2^{s}} \nonumber \\ =&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=0}^{|n|}\varepsilon _{s}\left( n\right) A_{n_{\left( s\right) }-1}^{\alpha _{n}}\left( D_{2^{s+1}}-D_{2^{s}}\right) \nonumber \\ =&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=1}^{|n|}\left( \varepsilon _{s-1}\left( n\right) -\varepsilon _{s}\left( n\right) \right) A_{n_{\left( s\right) }-1}^{\alpha _{n}}D_{2^{s}} + \nonumber \\&+ \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{s=1}^{|n|}\varepsilon _{s} \left( n\right) \left( A_{n_{\left( s-1\right) }-1}^{\alpha _{n}}-A_{n_{\left( s\right) }-1}^{\alpha _{n}}\right) D_{2^{s}} + \nonumber \\&+ \frac{1}{A_{n-1}^{\alpha _{n}}}\varepsilon _{|n|}\left( n\right) A_{n_{\left( |n|-1\right) }-1}^{\alpha _{n}}D_{2^{|n|+1}} - \nonumber \\&- \frac{1}{A_{n-1}^{\alpha _{n}}}\varepsilon _{0}\left( n\right) A_{n_{\left( 0\right) }-1}^{\alpha _{n}}D_{1} \nonumber \\ =:&Q_{31}(n) +Q_{32}(n) +Q_{33}(n) +Q_{34}(n). \end{aligned}$$

(3.6)

From (2.2)–(2.4) and (1.3) we get

$$\begin{aligned}{} & {} \left\| Q_{32}(n) \right\| _{1}\le \frac{1}{ A_{n-1}^{\alpha _{n}}}\sum \limits _{s=1}^{|n|}\left( A_{n_{\left( s\right) }-1}^{\alpha _{n}}-A_{n_{\left( s-1\right) }-1}^{\alpha _{n}}\right) \le c<\infty ,\nonumber \\{} & {} \left\| Q_{33}(n) \right\| _{1},\left\| Q_{34} (n) \right\| _{1}\le c<\infty , \end{aligned}$$

(3.7)

$$\begin{aligned}{} & {} \left\| Q_{31}(n) \right\| _{1}\le \frac{c}{n^{\alpha _{n}}}\sum \limits _{s=1}^{|n|}\left| \varepsilon _{s-1}\left( n\right) -\varepsilon _{s}\left( n\right) \right| 2^{s\alpha _{n}}. \end{aligned}$$

(3.8)

Combining (3.4)–(3.8) we conclude that

$$\begin{aligned} \left\| K_{n}^{\alpha _{n}}\right\| _{1}\le \frac{c_{1}}{n^{\alpha _{n}}}\sum \limits _{k=1}^{\left| n\right| }\left| \varepsilon _{k}\left( n\right) -\varepsilon _{k+1}\left( n\right) \right| 2^{k\alpha _{n}}. \end{aligned}$$

(3.9)

Hence the upper estimate is proved.

Now, we find a lower estimate for $\left\| Q_{31}(n) \right\| _{1}$. Let $a_{i}$ and $b_{i},i=1,\ldots ,s$, be strictly increasing sequences, i.e.,

$$\begin{aligned} 0\le a_{1}\le b_{1}<a_{2}\le b_{2}<\cdots<a_{s}\le b_{s}<a_{s+1}=\infty \end{aligned}$$

for which

$$\begin{aligned} \varepsilon _{j}(n) =\left\{ \begin{array}{ll} 1, &{} a_{i}\le j\le b_{i}, \\ 0, &{} b_{i}<j<a_{i+1}. \end{array} \right. \end{aligned}$$

(3.10)

Then it is evident that $b_{j}+1\le a_{j+1}$. But we may suppose even more, namely

$$\begin{aligned} b_{j}+2\le a_{j+1} \end{aligned}$$

(3.11)

and then the intervals defined below are disjoint sets. That is, set

$$\begin{aligned} A_{k}:=\left( \frac{1}{2^{a_{k}+1}},\frac{1}{2^{a_{k}}}\right) ,B_{k}:=\left( \frac{1}{2^{b_{k}+2}},\frac{1}{2^{b_{k}+1}}\right) ,k=1,\ldots ,s. \end{aligned}$$

Let $x\in A_{k}$. Then we can write

$$\begin{aligned} \left| Q_{31}(n)(x) \right|&= \left| \frac{-1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{s}A_{n_{ \left( a_{j}\right) }-1}^{\alpha _{n}}D_{2^{a_{j}}}\left( x\right) +\frac{1}{ A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{s}A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}D_{2^{b_{j}+1}}\left( x\right) \right| \nonumber \\&= \left| \frac{-1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{ \left( a_{j}\right) }-1}^{\alpha _{n}}2^{a_{j}}+\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k-1}A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}2^{b_{j}+1}\right| \nonumber \\&= \left| \frac{1}{A_{n-1}^{\alpha _{n}}}A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}2^{a_{k}}-\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=1}^{k-1}\left( A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}2^{b_{j}+1}-A_{n_{\left( a_{j}\right) }-1}^{\alpha _{n}}2^{a_{j}}\right) \right| . \end{aligned}$$

(3.12)

Since

$$\begin{aligned} b_{k-1}+1\le a_{k}-1<a_{k}, \end{aligned}$$

we can write

$$\begin{aligned}&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k-1}\left( A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}2^{b_{j}+1}-A_{n_{\left( a_{j}\right) }-1}^{\alpha _{n}}2^{a_{j}}\right) \\&\quad \le \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k-1}\left( A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}2^{b_{j}+1}-A_{n_{\left( b_{j-1}+1\right) }-1}^{\alpha _{n}}2^{b_{j-1}+1}\right) \\&\quad \le \frac{1}{A_{n-1}^{\alpha _{n}}}A_{n_{\left( b_{k-1}+1\right) }-1}^{\alpha _{n}}2^{b_{k-1}+1}. \end{aligned}$$

Consequently, from (3.12) we obtain

$$\begin{aligned} \left| Q_{31}(n) (x)\right|&\ge \frac{1}{ A_{n-1}^{\alpha _{n}}}\left( A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}2^{a_{k}}-A_{n_{\left( b_{k-1}+1\right) }-1}^{\alpha _{n}}2^{b_{k-1}+1}\right) \\&\ge \frac{1}{A_{n-1}^{\alpha _{n}}}\left( A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}2^{a_{k}}-A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}2^{a_{k}-1}\right) \\&= \frac{A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}}{A_{n-1}^{\alpha _{n}}} \left( 2^{a_{k}}-2^{a_{k}-1}\right) \\&= \frac{1}{2}\frac{A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}}{ A_{n-1}^{\alpha _{n}}}2^{a_{k}}. \end{aligned}$$

Hence,

$$\begin{aligned} \int \limits _{A_{k}}\left| Q_{31}(n) \right| \ge \frac{ A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}}{4}\frac{1}{A_{n-1}^{\alpha _{n}}}. \end{aligned}$$

(3.13)

Let $x\in B_{k}$. Since $n_{\left( b_{j}+1\right) }=n_{\left( b_{j}\right) }$ and $n_{\left( a_{j}\right) }\le n_{\left( b_{j}\right) }$, we have

$$\begin{aligned} w_{n}Q_{31}\left( n\right) (x)&= \frac{-1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=1}^{k}A_{n_{\left( a_{j}\right) }-1}^{\alpha _{n}}2^{a_{j}}+ \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{\left( b_{j}+1\right) }-1}^{\alpha _{n}}2^{b_{j}+1} \\&= \frac{-1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{\left( a_{j}\right) }-1}^{\alpha _{n}}2^{a_{j}}+\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{\left( b_{j}\right) }-1}^{\alpha _{n}}2^{b_{j}+1}\\&\ge \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{\left( b_{j}\right) }-1}^{\alpha _{n}}2^{b_{j}+1}-\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=1}^{k}A_{n_{\left( b_{j}\right) }-1}^{\alpha _{n}}2^{b_{j}} \\&\ge \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{k}A_{n_{\left( b_{j}\right) }-1}^{\alpha _{n}}2^{b_{j}} \\&\ge \frac{1}{A_{n-1}^{\alpha _{n}}}A_{n_{\left( b_{k}\right) }-1}^{\alpha _{n}}2^{b_{k}}. \end{aligned}$$

Hence

$$\begin{aligned} \int \limits _{B_{k}}\left| Q_{31}(n) \right| \ge \frac{ A_{n_{\left( b_{k}\right) }-1}^{\alpha _{n}}}{4}\frac{1}{A_{n-1}^{\alpha _{n}}}. \end{aligned}$$

(3.14)

Since $A_{i},B_{i}$, $i=1,\ldots ,s$, are pairwise disjoint from (3.13) and (3.14), we have

$$\begin{aligned} \int \limits _{\mathbb {I}}\left| Q_{31}(n) \right|&\ge \sum \limits _{k=1}^{s}\left( \int \limits _{A_{k}}\left| Q_{31}(n) \right| +\int \limits _{B_{k}}\left| Q_{31}(n) \right| \right) \\&\ge \frac{1}{4A_{n-1}^{\alpha _{n}}}\sum \limits _{k=1}^{s}\left( A_{n_{\left( a_{k}\right) }-1}^{\alpha _{n}}+A_{n_{\left( b_{k}\right) }-1}^{\alpha _{n}}\right) . \end{aligned}$$

Combining (3.5)–(3.14) we complete the proof of Theorem 3.1. $\square$

4 Uniform and L-convergence of $\left( C,\alpha _{n}\right)$ means

The Hardy space $H_{1}(\mathbb {I})$ consists all functions f that satisfy

$$\begin{aligned} \left\| f\right\| _{H_{1}}:=\left\| E^{*}\left( f\right) \right\| _{1}<\infty . \end{aligned}$$

For a nonnegative integer n let

$$\begin{aligned} V\left( n\right) :=\sum \limits _{i=0}^{\infty }\left| \varepsilon _{i}\left( n\right) -\varepsilon _{i+1}\left( n\right) \right| +\varepsilon _{0}\left( n\right) \end{aligned}$$

and

$$\begin{aligned} V(n,\alpha ) :=\frac{1}{n^{\alpha _{n}}}\sum \limits _{k=1}^{\left| n\right| }\left| \varepsilon _{k}(n) -\varepsilon _{k+1}(n) \right| 2^{k\alpha _{n}}. \end{aligned}$$

Let $C_{w}(\mathbb {I})$ denote the space of uniformly continuous functions on $\mathbb {I}$ with the supremum norm

$$\begin{aligned} \left\| f\right\| _{C_{w}}:=\sup \limits _{x\in \mathbb {I}}\left| f(x) \right| \ \ \ \ \left( f\in C_{w}(\mathbb { I}) \right) . \end{aligned}$$

Let $X=X(\mathbb {I})$ be either the space $L_{1}\left( \mathbb {I }\right)$, or the space of uniformly continuous functions, that is, $C_{w}(\mathbb {I})$. The corresponding norm is denoted by $\left\| \cdot \right\| _{X}$.

We remind the reader that $C_{w}(\mathbb {I})$ is the collection of all functions $f:\mathbb {I}\rightarrow \mathbb {R}$ that are uniformly continuous from the dyadic topology of $\mathbb {I}$ to the usual topology $\mathbb {R}$, or for short: uniformly w-continuous.

The modulus of continuity, when $X=C_{w}(\mathbb {I})$, and the integrated modulus of continuity, where $X=L_{1}(\mathbb {I})$ are defined by

$$\begin{aligned} \omega \left( \frac{1}{2^{n}},f\right) _{X}=\sup \limits _{h\in I_{n}}\left\| f( \cdot \dotplus h) -f( \cdot ) \right\| _{X}. \end{aligned}$$

For Walsh–Fourier series Fine [5] has obtained a sufficient condition for the uniform convergence which is in complete analogy with the Dini–Lipshitz condition (see also [23]). Similar results are true for the space of integrable functions $L_{1}(\mathbb {I})$ [22]. Gulicev [16] has estimated the rate of uniform convergence of a Walsh–Fourier series using the Lebesgue constant and the modulus of continuity. Uniform convergence of Walsh–Fourier series of the functions of classes of generalized bounded variation was investigated by the second author [13]. This problem has been considered for the Vilenkin group by Fridli [7] and Gát [9]. Lukomskii [19] considered uniform and $L_{1}$ -convergence of subsequences of partial sums of Walsh–Fourier series. In particular, he proved that the condition $\sup \limits _{A}V( m_{A}) <\infty$ is necessary and sufficient for the uniform and $L_{1}$-convergence of subsequences of partial sums $S_{m_{A}}(f)$ of Walsh–Fourier series. In [4, 10,11,12, 15, 18,19,20,21] the X -norm convergence of subsequences of Walsh–Fourier series is investigated.

In this section we discuss some convergence and divergence properties of subsequences of Cesàro means with varying parameters of Walsh–Fourier series. The following are true.

Theorem 4.1

If $f\in X(\mathbb {I})$ and $\alpha _{n}\in \left( 0,1\right)$, then

$$\begin{aligned} \left\| \sigma _{n}^{\alpha _{n}}\left( f\right) -f\right\| _{X} \le&c_{1}\omega \left( \frac{1}{2^{|n|}},f\right) _{X}V(n,\alpha ) \\&+ c_{2}\alpha _{n}\sum \limits _{r=0}^{|n|-2}2^{r-|n|}\omega \left( \frac{1}{ 2^{r}},f\right) _{X} \\&+ c_{3}\omega \left( \frac{1}{2^{|n|-1}},f\right) _{X}. \end{aligned}$$

Theorem 4.1 implies

Theorem 4.2

If for a function $f\in X(\mathbb {I})$ and a subsequence $\left\{ m_{n}:n\in \mathbb {N}\right\}$ we have

$$\begin{aligned} \omega \left( \frac{1}{m_{n}},f\right) _{X}=o\left( \frac{1}{V( m_{n},\alpha ) }\right) , \end{aligned}$$

then the subsequence $\sigma _{m_{n}}^{\alpha _{n}}(f)$ converges in X-norm.

Theorem 4.3

Let $\left\{ m_{n}:n\in \mathbb {N}\right\}$ be such that

$$\begin{aligned} \sup \limits _{n}V(m_{n},\alpha ) =\infty . \end{aligned}$$

Then there exists $\left\{ p_{k}:k\in \mathbb {N}\right\} \$and a function $g\in X(\mathbb {I})$ such that

$$\begin{aligned} \omega \left( \frac{1}{m_{p_{k}}},g\right) _{X}=O\left( \frac{1}{V(m_{p_{k}},\alpha ) }\right) \end{aligned}$$

and

$$\begin{aligned} \left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g\right) -g\right\| _{1}\nrightarrow 0\text { as }k\rightarrow \infty \text {.} \end{aligned}$$

Theorem 4.4

a) Let $\left\{ m_{n}:n\in \mathbb {N}\right\}$ be such that

$$\begin{aligned} \sup \limits _{n}V(m_{n},\alpha ) <\infty . \end{aligned}$$

Then the operator $\sigma _{m_{n}}^{\alpha _{n}}(f)$ is bounded from the space $L_{1}(\mathbb {I})$ to the space $L_{1}(\mathbb {I})$;b) Let $\left\{ m_{n}:n\in \mathbb {N}\right\}$ be such that

$$\begin{aligned} \sup \limits _{n}V(m_{n},\alpha ) =\infty . \end{aligned}$$

Then there exists $f\in H_{1}(\mathbb {I})\subset L_{1}( \mathbb {I})$ such that

$$\begin{aligned} \sup \limits _{n}\left\| \sigma _{m_{n}}^{\alpha _{n}}(f) \right\| _{1}=\infty . \end{aligned}$$

Proof of Theorem 4.1

We can write

$$\begin{aligned}&\sigma _{n}^{\alpha _{n}}\left( x,f\right) -f(x) \nonumber \\&\quad = \int \limits _{\mathbb {I}}\left[ f\left( x\dotplus t\right) -f\left( x\right) \right] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&\quad = \int \limits _{\mathbb {I}}\left[ f\left( x\dotplus t\right) -f\left( x\right) \right] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=1}^{2^{|n|-1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&\qquad + \int \limits _{\mathbb {I}}\left[ f\left( x\dotplus t\right) -f\left( x\right) \right] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|-1}}^{2^{|n|}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&\qquad + \int \limits _{\mathbb {I}}\left[ f\left( x\dotplus t\right) -f\left( x\right) \right] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&\quad =: J_{1}+J_{2}+J_{3}. \end{aligned}$$

(4.1)

For $J_{3}$ we have

$$\begin{aligned} J_{3} =&\int \limits _{\mathbb {I}}[f(x\dotplus t) -S_{2^{|n|}}(x\dotplus t,f)] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&+ \int \limits _{\mathbb {I}}[ S_{2^{|n|}}(x\dotplus t,f) -S_{2^{|n|}}(x,f)] \frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&+ \int \limits _{\mathbb {I}}\left[ S_{2^{|n|}}\left( x,f\right) -f\left( x\right) \right] \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\ =:&J_{31}+J_{32}+J_{33}. \end{aligned}$$

(4.2)

Now we can write

$$\begin{aligned} J_{32}&= \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}\left( S_{j}*S_{2^{|n|}}\right) (x;f) -\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}S_{2^{|n|}}\left( x;f\right) \nonumber \\&= \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}S_{2^{|n|}}\left( x;f\right) -\frac{1}{A_{n-1}^{\alpha _{n}}} \sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}S_{2^{|n|}}\left( x;f\right) = 0. \end{aligned}$$

(4.3)

From Minkowski’s inequality and Theorem 3.1 we get

$$\begin{aligned} \left\| J_{31}\right\| _{X}&\le \omega \left( \frac{1}{2^{|n|}} ,f\right) _{X}\int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}} \left| \sum \limits _{j=2^{|n|}}^{n}A_{n-j}^{\alpha _{n}-1}D_{j}\left( t\right) \right| \,dt \nonumber \\&\le \omega \left( \frac{1}{2^{|n|}},f\right) _{X}\left( 1+V( n,\alpha ) \right) . \end{aligned}$$

(4.4)

Analogously, we can prove

$$\begin{aligned} \left\| J_{33}\right\| _{X}\le \omega \left( \frac{1}{2^{|n|}} ,f\right) _{X}\left( 1+V\left( n,\alpha \right) \right) . \end{aligned}$$

(4.5)

Combining (4.2)–(4.5) we obtain

$$\begin{aligned} \left\| J_{3}\right\| _{X}\le 2\omega \left( \frac{1}{2^{|n|}} ,f\right) _{X}\left( 1+V(n,\alpha ) \right) . \end{aligned}$$

(4.6)

We can write

$$\begin{aligned} J_{1} =&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{r=0}^{|n|-2}\int \limits _{\mathbb {I}}\left[ f\left( x\dotplus t\right) -f(x) \right] \sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}\left( t\right) \,dt \nonumber \\ =&\frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{r=0}^{|n|-2}\int \limits _{ \mathbb {I}}[f(x\dotplus t) -S_{2^{r}}(x\dotplus t,f)] \sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\&+ \frac{1}{A_{n-1}^{\alpha _{n}}}\sum \limits _{r=0}^{|n|-2}\int \limits _{ \mathbb {I}}\left[ S_{2^{r}}\left( x,f\right) -f(x) \right] \sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \,dt \nonumber \\ =:&J_{11}+J_{12}. \end{aligned}$$

(4.7)

Using Minkowski’s inequality we get

$$\begin{aligned} \left\| J_{11}\right\| _{X}\le \sum \limits _{r=0}^{|n|-2}\omega \left( \frac{1}{2^{r}},f\right) _{X}\frac{1}{A_{n-1}^{\alpha _{n}}}\int \limits _{ \mathbb {I}}\left| \sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \right| \,dt . \end{aligned}$$

Since

$$\begin{aligned}&\sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}\left( t\right) \\&\quad = \sum \limits _{j=2^{r}}^{2^{r+1}-2}A_{n-j}^{\alpha _{n}-2}jK_{j}\left( t\right) +A_{n-2^{r+1}+1}^{\alpha _{n}-1}\left( 2^{r+1}-1\right) K_{2^{r+1}-1}\left( t\right) \\&\qquad - A_{n-2^{r}}^{\alpha _{n}-1}\left( 2^{r}-1\right) K_{2^{r}-1}\left( t\right) \end{aligned}$$

and

$$\begin{aligned} \sup \limits _{n}\left\| K_{n}\right\| _{1}<2, \end{aligned}$$

from (2.2)–(2.4) we get $\left( r<|n|-1\right)$

$$\begin{aligned}&\frac{1}{A_{n-1}^{\alpha _{n}}}\int \limits _{\mathbb {I}}\left| \sum \limits _{j=2^{r}}^{2^{r+1}-1}A_{n-j}^{\alpha _{n}-1}D_{j}(t) \right| \,dt \\&\quad \le \frac{2}{A_{n-1}^{\alpha _{n}}}\left\{ \sum \limits _{j=2^{r}}^{2^{r+1}-2}\left| A_{n-j}^{\alpha _{n}-2}\right| j+A_{n-2^{r+1}+1}^{\alpha _{n}-1}\left( 2^{r+1}-1\right) +A_{n-2^{r}}^{\alpha _{n}-1}\left( 2^{r}-1\right) \right\} \\&\quad \le \frac{c\left( 1-\alpha _{n}\right) }{n^{\alpha _{n}}}\Bigg \{ \sum \limits _{j=2^{r}}^{2^{r+1}-2}\left( n-j\right) ^{\alpha _{n}-2}j+\left( n-2^{r+1}+1\right) ^{\alpha _{n}-1}\left( 2^{r+1}-1\right) \\&\qquad +\left( n-2^{r}\right) ^{\alpha _{n}-1}\left( 2^{r}-1\right) \Bigg \} \\&\quad \le \frac{c\left( 1-\alpha _{n}\right) \alpha _{n}}{n^{\alpha _{n}}}\frac{ 2^{2r}}{\left( n-2^{\left| n\right| -1}\right) ^{2-\alpha _{n}}}+ \frac{c2^{r}\left( 1-\alpha _{n}\right) \alpha _{n}}{n^{\alpha _{n}}\left( n-2^{\left| n\right| -1}\right) ^{1-\alpha _{n}}} \\&\quad \le c\alpha _{n}2^{r-|n|}. \end{aligned}$$

Consequently,

$$\begin{aligned} \left\| J_{11}\right\| _{X}\le c\alpha _{n}\sum \limits _{r=0}^{|n|-2}2^{r-|n|}\omega \left( \frac{1}{2^{r}},f\right) . \end{aligned}$$

(4.8)

Analogously, we can prove

$$\begin{aligned} \left\| J_{12}\right\| _{X}\le c\alpha _{n}\sum \limits _{r=0}^{|n|-2}2^{r-|n|}\omega \left( \frac{1}{2^{r}},f\right) . \end{aligned}$$

(4.9)

Combining (4.7)–(4.9) we get

$$\begin{aligned} \left\| J_{1}\right\| _{X}\le c\alpha _{n}\sum \limits _{r=0}^{|n|-2}2^{r-|n|}\omega \left( \frac{1}{2^{r}},f\right) . \end{aligned}$$

(4.10)

Finally, we estimate $J_{2}$. Since $D_{2^{m}-j}=D_{2^{m}}-w_{2^{m}-1}D_{j},j=0,1,\ldots ,2^{m}-1$, we have

$$\begin{aligned}&\int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}}\left| \sum \limits _{j=2^{|n|-1}}^{2^{|n|}-1}A_{n-j}^{\alpha _{n}-1}D_{j}\left( t\right) \right| \,dt \\&\quad = \int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}}\left| \sum \limits _{j=1}^{2^{|n|-1}}A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-1}D_{2^{|n|}-j}\left( t\right) \right| \,dt \\&\quad \le \int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}}\left| \sum \limits _{j=1}^{2^{|n|-1}}A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-1}D_{2^{|n|}}\left( t\right) \right| \,dt \\&\qquad + \int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}}\left| A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-1}D_{j}(t) \right| \,dt \\&\quad \le c_{1}+\int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}} \left| A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-1}D_{j}\left( t\right) \right| \,dt \\&\quad \le c_{1}+\int \limits _{\mathbb {I}}\frac{1}{A_{n-1}^{\alpha _{n}}} \left| \sum \limits _{j=1}^{2^{|n|-1}-1}A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-2}jK_{j}\left( t\right) \right| \,dt \\&\qquad + \int \limits _{\mathbb {I}}\frac{A_{n_{\left( |n|-1\right) }+2^{|n|-1}}^{\alpha _{n}-1}}{A_{n-1}^{\alpha _{n}}}2^{|n|-1}\left| K_{2^{|n|-1}}\left( t\right) \right| \,dt \\&\quad \le c_{1}+\frac{c_{2}\sum \limits _{j=0}^{2^{|n|-1}} A_{n_{\left( |n|-1\right) }+j}^{\alpha _{n}-1}}{A_{n-1}^{\alpha _{n}}}\le c<\infty . \end{aligned}$$

Hence,

$$\begin{aligned} \left\| J_{2}\right\| _{X}\le c\omega \left( \frac{1}{2^{|n|-1}} ,f\right) _{X}. \end{aligned}$$

(4.11)

Combining (4.1), (4.6), (4.10) and (4.11) we complete the proof of Theorem 4.1. $\square$

Proof of Theorem 4.3

Since $\sup \limits _{n}V(m_{n},\alpha ) =\infty$, there exists $\left\{ p_{k}:k\in \mathbb {N}\right\}$ such that

$$\begin{aligned}{} & {} \left| m_{p_{k}}\right| >\left| m_{p_{k-1}}\right| +2\log (k+1); \end{aligned}$$

(4.12)

$$\begin{aligned}{} & {} V(m_{p_{k}},\alpha ) \ge 2kV(m_{p_{k-1}},\alpha ). \end{aligned}$$

(4.13)

At first we consider the case $X(\mathbb {I})=L_{1}\left( \mathbb {I}\right)$. We set

$$\begin{aligned} g\left( x\right) :=\sum \limits _{j=1}^{\infty }g_{j}(x), \ g_{j}(x) :=\frac{D_{2^{|m_{p_{j}}|+1}}(x) }{V( m_{p_{j}},\alpha )}. \end{aligned}$$

If $y\in I_{|m_{p_{k}}|}$, then for $l=1,2,\ldots ,k-1$ we obtain

$$\begin{aligned} D_{2^{|m_{p_{l}}|}}\left( x\dotplus y\right) =D_{2^{|m_{p_{l}}|}}\left( x\right) . \end{aligned}$$

(4.14)

Then, from (1.3) and (4.13),

$$\begin{aligned}&\int \limits _{\mathbb {I}}\left| g\left( x\dotplus y\right) -g\left( x\right) \right| dx \\&\quad \le \sum \limits _{j=k}^{\infty }\frac{1}{V\left( m_{p_{j}},\alpha \right) } \int \limits _{\mathbb {I}}\left| D_{2^{|m_{p_{j}}|+1}}\left( x\dotplus y\right) -D_{2^{|m_{p_{j}}|+1}}\left( x\right) \right| dx \\&\quad \le \frac{c}{V\left( m_{p_{k}},\alpha \right) }. \end{aligned}$$

Consequently,

$$\begin{aligned} \omega \left( \frac{1}{m_{p_{k}}},g\right) _{1}=O\left( \frac{1}{V\left( m_{p_{p}},\alpha \right) }\right) . \end{aligned}$$

Further,

$$\begin{aligned}&\left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g\right) -g\right\| _{1} \nonumber \\&\quad \ge \left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( \sum \limits _{j=k}^{\infty }g_{j}\right) \right\| _{1}-\sum \limits _{j=k}^{\infty }\left\| g_{j}\right\| _{1} - \left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( \sum \limits _{j=1}^{k-1}g_{j}\right) -\sum \limits _{j=1}^{k-1}g_{j}\right\| _{1}. \end{aligned}$$

(4.15)

Since for $j\ge k$

$$\begin{aligned} \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g_{j}\right)&= g_{j}*K_{m_{p_{k}}}^{\alpha _{p_{k}}}=K_{m_{p_{k}}}^{\alpha _{p_{k}}}*g_{j}= \frac{1}{V\left( m_{p_{j}},\alpha \right) }K_{m_{p_{k}}}^{\alpha _{p_{k}}}*D_{2^{|m_{p_{j}}|+1}} \\&= \frac{1}{V\left( m_{p_{j}},\alpha \right) }S_{2^{|m_{p_{j}}|+1}}\left( K_{m_{p_{k}}}^{\alpha _{p_{k}}}\right) =\frac{1}{V\left( m_{p_{j}},\alpha \right) }K_{m_{p_{k}}}^{\alpha _{p_{k}}}, \end{aligned}$$

from Theorem 3.1 we obtain

$$\begin{aligned} \left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( \sum \limits _{j=k}^{\infty }g_{j}\right) \right\| _{1}=\sum \limits _{j=k}^{\infty }\frac{1}{V\left( m_{p_{j}},\alpha \right) } \left\| K_{m_{p_{k}}}^{\alpha _{p_{k}}}\right\| _{1}\ge \frac{ \left\| K_{m_{p_{k}}}^{\alpha _{p_{k}}}\right\| _{1}}{V\left( m_{p_{k}},\alpha \right) }\ge c_{0}>0. \end{aligned}$$

(4.16)

From (1.3) and (4.13) we get

$$\begin{aligned} \sum \limits _{j=k}^{\infty }\left\| g_{j}\right\| _{1}\le 2\sum \limits _{j=k}^{\infty }\frac{1}{V\left( m_{p_{j}},\alpha \right) }\le \frac{3}{V\left( m_{p_{k}},\alpha \right) }. \end{aligned}$$

(4.17)

Using Theorem 4.1, from (4.12) and (4.14) we obtain for $j<k$

$$\begin{aligned} \left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g_{j}\right) -g_{j}\right\|&\le c\alpha _{p_{k}}\sum \limits _{r=0}^{|m_{p_{k}}|-2}2^{r-|m_{p_{k}}|}\omega \left( g_{j},\frac{1}{2^{r}}\right) _{1} \\&\le c\alpha _{p_{k}}\sum \limits _{r=0}^{|m_{p_{j}}|-2}2^{r-|m_{p_{k}}|}\omega \left( g_{j},\frac{1}{2^{r}}\right) _{1} \\&\le \frac{c}{2^{|m_{p_{k}}|}}\sum \limits _{r=0}^{|m_{p_{j}}|}2^{r} \\&\le \frac{c}{k^{2}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \sum \limits _{j=1}^{k-1}\left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g_{j}\right) -g_{j}\right\| _{1}\le \frac{c}{k}. \end{aligned}$$

(4.18)

Combining (4.15)–(4.18) we obtain

$$\begin{aligned} \overline{\lim \limits _{k\rightarrow \infty }}\left\| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( g\right) -g\right\| _{1}>0. \end{aligned}$$

Now, we discuss the case $X(\mathbb {I})=C_{w}\left( \mathbb {I} \right)$. Let the conditions (4.12) and (4.13) be satisfied. We set

$$\begin{aligned} h(x) :=\sum \limits _{j=1}^{\infty }\frac{h_{j}(x) }{ V(m_{p_{j}},\alpha )}, \end{aligned}$$

where

$$\begin{aligned} h_{j}(x) :=\text {sgn}\left( K_{m_{p_{j}}}^{\alpha _{p_{j}}}\right) . \end{aligned}$$

It is evident that $h_{0}\in C_{w}(\mathbb {I})$. If $y\in I_{|m_{p_{k}}|}$, then for $j=1,2,\ldots ,k-1$ we obtain

$$\begin{aligned} h_{j}\left( x\dotplus y\right) -h_{j}\left( x\right) =0 \end{aligned}$$

and from (4.13) we get

$$\begin{aligned} \left| h\left( x\dotplus y\right) -h\left( x\right) \right| \le 2\sum \limits _{j=k}^{\infty }\frac{1}{V\left( m_{p_{j}},\alpha \right) } =O\left( \frac{1}{V\left( m_{p_{k}},\alpha \right) }\right) . \end{aligned}$$

Hence

$$\begin{aligned} \omega \left( \frac{1}{m_{p_{k}}},h\right) _{C}=O\left( \frac{1}{V\left( m_{p_{k}},\alpha \right) }\right) . \end{aligned}$$

Clearly

$$\begin{aligned}&\left| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( 0,h\right) -h\left( 0\right) \right| \nonumber \\&\quad \ge \frac{\left| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( 0,h_{k}\right) \right| }{V\left( m_{p_{k}},\alpha \right) } -\sum \limits _{j=k}^{\infty }\frac{\left| h_{j}\left( 0\right) \right| }{V\left( m_{p_{j}},\alpha \right) }-\sum \limits _{j=k+1}^{\infty }\frac{\left| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( 0,h_{j}\right) \right| }{V\left( m_{p_{j}},\alpha \right) } \nonumber \\&\qquad - \sum \limits _{j=1}^{k-1}\frac{\left| \sigma _{m_{p_{k}}}^{\alpha _{p_{k}}}\left( 0,h_{j}\right) -h_{j}\left( 0\right) \right| }{V\left( m_{p_{j}},\alpha \right) } \nonumber \\&\quad =: R_{1}-R_{2}-R_{3}-R_{4}. \end{aligned}$$

(4.19)

Using Theorem 3.1, from (4.12) and (4.13) we obtain

$$\begin{aligned} R_{1}= & {} \frac{\left\| K_{m_{p_{k}}}^{\alpha _{p_{k}}}\right\| _{1}}{ V\left( m_{p_{k}},\alpha \right) }\ge c>0, \end{aligned}$$

(4.20)

$$\begin{aligned} R_{2}\le & {} \frac{c}{V\left( m_{p_{k}},\alpha \right) }, \end{aligned}$$

(4.21)

$$\begin{aligned} R_{3}\le & {} \sum \limits _{j=k+1}^{\infty }\frac{\left\| K_{m_{p_{k}}}^{\alpha _{p_{k}}}\right\| _{1}}{V\left( m_{p_{j}},\alpha \right) }\le \frac{cV\left( m_{p_{k}},\alpha \right) }{V\left( m_{p_{k+1}},\alpha \right) }\le \frac{c}{k}, \end{aligned}$$

(4.22)

$$\begin{aligned} R_{4}\le & {} c\sum \limits _{j=1}^{k-1}\sum \limits _{r=0}^{|m_{p_{k}}|-2}2^{r-|m_{p_{k}}|} \omega \left( \frac{1}{2^{r}},h_{j}\right) _{C} \nonumber \\{} & {} \le c\sum \limits _{j=1}^{k-1}\sum \limits _{r=0}^{|m_{p_{j}}|}2^{r-|m_{p_{k}}|} \omega \left( \frac{1}{2^{r}},h_{j}\right) _{C} \nonumber \\{} & {} \le \frac{ck2^{|m_{p_{k-1}}|}}{2^{|m_{p_{k}}|}}\le \frac{c}{k}. \end{aligned}$$

(4.23)

Combining (4.19)–(4.23) we complete the proof of Theorem 4.3. $\square$

Proof of Theorem 4.4

The validity of part a) immediately follows from Theorem 3.1. Now, we prove part b). Since $\sup \limits _{a}V\left( m_{a},\alpha \right) =\infty$, without loss of generality we can suppose

$$\begin{aligned} V\left( m_{a},\alpha \right) \ge a^{4}. \end{aligned}$$

(4.24)

Set

$$\begin{aligned} f:=\sum \limits _{a=1}^{\infty }\lambda _{a}f_{a},, \end{aligned}$$

where

$$\begin{aligned} \lambda _{a}:=\frac{1}{\sqrt{V\left( m_{a},\alpha \right) }} \end{aligned}$$

and

$$\begin{aligned} f_{a}:=D_{2^{|m_{a}|+1}}-D_{2^{|m_{a}|}}. \end{aligned}$$

We can write

$$\begin{aligned} S_{2^{n}}\left( f_{a}\right) =S_{2^{n}}\left( D_{2^{|m_{a}|+1}}-D_{2^{|m_{a}|}}\right) =\left\{ \begin{array}{ll} 0, &{} \text { if }n\le |m_{a}|, \\ f_{a}, &{}\text { if }n>|m_{a}|. \end{array} \right. \end{aligned}$$

Hence,

$$\begin{aligned} S_{2^{n}}\left( f\right) =\sum \limits _{a=1}^{\infty }\lambda _{a}S_{2^{n}}\left( f_{a}\right) =\sum \limits _{\left\{ a : |m_{a}|<n\right\} }\lambda _{a}f_{a} \end{aligned}$$

and

$$\begin{aligned} \sup \limits _{n}\left| S_{2^{n}}\left( f\right) \right| \le \sum \limits _{a}\lambda _{a}\left| f_{a}\right| =\sum \limits _{a}\lambda _{a}D_{2^{|m_{a}|}}. \end{aligned}$$

Applying (1.3) and (4.24) we conclude that

$$\begin{aligned} \left\| \sup \limits _{n}\left| S_{2^{n}}\left( f\right) \right| \right\| _{1}\le \sum \limits _{a}\lambda _{a}\le \sum \limits _{a}\frac{1}{ a^{2}}<\infty . \end{aligned}$$

Hence $f\in H_{1}(\mathbb {I})$.

We can write

$$\begin{aligned} \sigma _{m_{a}}^{\alpha _{a}}\left( f\right) =\sigma _{m_{a}}^{\alpha _{a}}\left( \lambda _{a}f_{a}\right) +\sum \limits _{j=0}^{a-1}\lambda _{j}\sigma _{m_{a}}^{\alpha _{a}}\left( f_{j}\right) +\sum \limits _{j=a+1}^{\infty }\lambda _{j}\sigma _{m_{a}}^{\alpha _{a}}\left( f_{j}\right) . \end{aligned}$$

(4.25)

Let $j>a$. Then it is easy to see that

$$\begin{aligned} \sigma _{m_{a}}^{\alpha _{a}}\left( f_{j}\right)&= f_{j}*K_{m_{a}}^{\alpha _{a}}=\left( D_{2^{|m_{j}|+1}}-D_{2^{|m_{j}|}}\right) *K_{m_{a}}^{\alpha _{a}} \nonumber \\&= S_{2^{|m_{j}|+1}}\left( K_{m_{a}}^{\alpha _{a}}\right) -S_{2^{|m_{j}|}}\left( K_{m_{a}}^{\alpha _{a}}\right) =0. \end{aligned}$$

(4.26)

Let $j<a$. Then from Theorem 4.1 we have

$$\begin{aligned} \left\| \sigma _{m_{a}}^{\alpha _{a}}\left( f_{j}\right) \right\| _{1}&\le \left\| \sigma _{m_{a}}^{\alpha _{a}}\left( f_{j}\right) -f_{j}\right\| _{1}+2 \nonumber \\&\le \sum \limits _{r=0}^{|m_{a}|}2^{r-|m_{a}|}\omega \left( f_{j},\frac{1}{2^{r}}\right) _{1}+2 \nonumber \\&\le \sum \limits _{r=0}^{|m_{j}|}2^{r-|m_{a}|}\omega \left( f_{j},\frac{1}{2^{r}}\right) _{1}+2 \nonumber \\&\le \frac{2^{|m_{a-1}|}}{2^{|m_{a}|}}+2<3. \end{aligned}$$

(4.27)

Finally, we estimate $\sigma _{m_{a}}^{\alpha _{a}}\left( f_{a}\right)$. Let $m_{a}=2^{|m_{a}|}+m_{a}^{\prime }, 0\le m_{a}^{\prime }<2^{|m_{a}|}$. Then we can write

$$\begin{aligned} \left| \sigma _{m_{a}}^{\alpha _{a}}\left( f_{a}\right) \right| = \frac{1}{A_{m_{a}-1}^{\alpha _{a}}}\left| \sum \limits _{j=0}^{m_{a}^{\prime }-1}A_{m_{a}^{\prime }-1-j}^{\alpha _{a}}w_{j}\right| \end{aligned}$$

and consequently, from Theorem 3.1 and by (2.2)–(2.4), we get

$$\begin{aligned} \left\| \sigma _{m_{a}}^{\alpha _{a}}\left( f_{a}\right) \right\| _{1}&= \frac{1}{A_{m_{a}-1}^{\alpha _{a}}}\left\| \sum \limits _{j=0}^{m_{a}^{\prime }-1}A_{m_{a}^{\prime }-1-j}^{\alpha _{a}}w_{j}\right\| _{1} \nonumber \\&\ge \frac{c}{A_{m_{a}-1}^{\alpha _{a}}}\sum \limits _{j=0}^{|m_{a}^{\prime }|}\left| \varepsilon _{j}\left( m_{a}^{\prime }\right) -\varepsilon _{j+1}\left( m_{a}^{\prime }\right) \right| 2^{j\alpha _{a}} \nonumber \\&\ge \frac{c}{2^{|m_{a}|\alpha _{a}}}\sum \limits _{j=0}^{|m_{a}|}\left| \varepsilon _{j}\left( m_{a}\right) -\varepsilon _{j+1}\left( m_{a}\right) \right| 2^{j\alpha _{a}} \nonumber \\&\ge cV\left( m_{a},\alpha \right) . \end{aligned}$$

(4.28)

Combining (4.25)–(4.28) we conclude that

$$\begin{aligned} \sup \limits _{a}\left\| \sigma _{m_{a}}^{\alpha _{a}}\left( f\right) \right\| _{1}=\infty . \end{aligned}$$

$\square$