On the strong convergence of partial sums with respect to bounded Vilenkin systems

TL;DR

This paper studies the strong convergence behavior of partial sums in Vilenkin systems, providing new theorems that enhance understanding of their convergence properties in harmonic analysis.

Contribution

It introduces novel strong convergence theorems for partial sums in Vilenkin systems, advancing theoretical knowledge in this area.

Findings

Established new strong convergence theorems for Vilenkin partial sums

Enhanced understanding of convergence properties in harmonic analysis

Provided conditions under which convergence is guaranteed

Abstract

In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.

Equations

$${\Vert S_{M_k}f\Vert }_{H_1}\le c{\Vert f\Vert }_{H_1}(k\in \mathbb{N}).$$

$${\Vert f\Vert }_{H_1}\equiv {\Vert \underset{n}{\sup }\left|S_{M_n}f\right|\Vert }_1.$$

$$\underset{n\to \infty }{\lim }\frac{1}{\log n}\stackrel{n}{\sum _{k=1}}\frac{{\Vert S_{k}f-f\Vert }_1}{k}=0,$$

$$\frac{1}{\log n}\stackrel{n}{\sum _{k=1}}\frac{{\Vert S_{k}f\Vert }_1}{k}\le c{\Vert f\Vert }_{H_1}\left(n=2,\mathrm{3...}\right)$$

$$\underset{n\to \infty }{\lim }\frac{1}{\log n}\stackrel{n}{\sum _{k=1}}\frac{{\Vert S_{k}f\Vert }_1}{k}={\Vert f\Vert }_{H_1},$$

$$\underset{n\in {\mathbb{N}}_{+}}{\sup }\frac{1}{n}\underset{m=1}{\sum ^n}{\Vert S_{m}f\Vert }_1\le c{\Vert f\Vert }_{H_1}.$$

$$\underset{n\in {\mathbb{N}}_{+}}{\sup }{\Vert \frac{1}{n}\underset{m=1}{\sum ^n}{S}_{m}f\Vert }_1<c{\Vert f\Vert }_{H_1}.$$

$$\underset{n\in {\mathbb{N}}_{+}}{\sup }\frac{1}{n}\underset{m=1}{\sum ^n}{\Vert S_{m}f\Vert }_1=\infty .$$

$$Z_{m_k}:=\{0,1,\dots ,m_k-1\}$$

$${\mu }_k\left(\{j\}\right):=1/m_k(j\in Z_{m_k})$$

$$x:=(x_0,x_1,\dots ,x_k,\dots )\left(x_k\in Z_{m_k}\right).$$

$$I_0\left(x\right):=G_m,$$

$$I_n(x):=\{y\in G_m\mid y_0=x_0,\dots ,y_{n-1}=x_{n-1}\}(x\in G_m,n\in \mathbb{N})$$

$$e_n:=\left(0,\dots ,0,x_n=1,0,\dots \right)\in G_m\left(n\in \mathbb{N}\right).$$

$$M_0:=1,M_{k+1}:=m_k{M}_k(k\in \mathbb{N})$$

$${\delta }_j=sign{n}_j=sign\left(\ominus n_j\right),{\delta }_j^{\ast }=\left|\ominus n_j-1\right|{\delta }_j,$$

$$v\left(n\right)=\sum _{j=0}^{\infty }\left|{\delta }_{j+1}-{\delta }_j\right|+{\delta }_0,v^{\ast }\left(n\right)=\sum _{j=0}^{\infty }{\delta }_j^{\ast },$$

$$r_k\left(x\right):=\exp \left(2\pi ıx_k/m_k\right)\left({ı}^2=-1,x\in G_m,k\in \mathbb{N}\right).$$

$${\psi }_n\left(x\right):=\prod _{k=0}^{\infty }{r}_k^{n_k}\left(x\right)\left(n\in \mathbb{N}\right).$$

$${\Vert f\Vert }_p:={\left({\int }_{G_m}{\left|f(x)\right|}^{p}d\mu (x)\right)}^{1/p}\left(0<p<\infty \right).$$

$$\hat{f}(k)$$

$$S_{n}f$$

$${\sigma }_{n}f$$

$$D_n$$

$$\quad D_{M_{n}}\left(x\right)=\left\{\begin{array}[]{l}\text{ }M_{n}\text{ \ \ \ }x\in I_{n}\\ \text{ }0\text{ \qquad}x\notin I_{n}\end{array}\right.$$

$$D_{s_n{M}_n}=D_{M_n}\sum _{k=0}^{s_n-1}{\psi }_{k{M}_n}=D_{M_n}\sum _{k=0}^{s_n-1}{r}_n^{k}1\le s_n\le m_n-1.$$

$$L_n={\Vert D_n\Vert }_1.$$

$$f^{\ast }\left(x\right)=\underset{n\in \mathbb{N}}{\sup }\frac{1}{\left|I_n\left(x\right)\right|}\left|{\int }_{I_n\left(x\right)}f\left(u\right)\mu \left(u\right)\right|$$

$${\Vert f\Vert }_{H_1}:={\Vert f^{\ast }\Vert }_1<\infty .$$

$$\frac{1}{4\lambda }v\left(n\right)+\frac{1}{\lambda }{v}^{\ast }\left(n\right)+\frac{1}{2\lambda }\le L_n\le \frac{3}{2}v\left(n\right)+4v^{\ast }\left(n\right)-1,$$

$$\frac{1}{n{M}_n}\underset{k=1}{\sum ^{M_n-1}}v\left(k\right)\ge c>0.$$

$$\underset{n\in \mathbb{N}}{\sup }\frac{1}{n}\stackrel{n}{\sum _{k=1}}{\Vert S_{k}f\Vert }_1=\infty .$$

Full text

The research was supported by a Swedish Institute scholarship and by Shota Rustaveli National Science Foundation grant YS15-2.1.1-47.

2010 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, partial sums, Fejér means, martingale Hardy space, strong convergence.

1. Introduction

It is well-known (for details see e.g. [10] and [14]) that Vilenkin system does not form basis in the space $L_{1}\left(G_{m}\right).$ Moreover, there is a function in the Hardy space $H_{1}\left(G_{m}\right),$ such that the partial sums of $f$  are not bounded in $L_{1}$-norm. However, subsequence $S_{M_{n}}$ of partial sums are bounded from the martingale Hardy space $H_{1}\left(G_{m}\right)$ to the Lebesgue space $L_{1}\left(G_{m}\right):$

[TABLE]

Moreover, we have the follwing norm equivalence:

[TABLE]

Moreover, Gát [8] proved the following strong convergence result for all $f\in H_{1}:$

[TABLE]

where $S_{k}f$ denotes the $k$-th partial sum of the Vilenkin-Fourier series of $f.$

It follows that there exists an absolute constant $c,$ such that

[TABLE]

and

[TABLE]

for all $f\in H_{1}.$

Analogical result for the trigonometric system was proved by Smith [20], for the Walsh-Paley system by Simon [18].

If partial sums of Vilenkin-Fourier series was bounded from $H_{1}$ to $L_{1}$ we also would have:

[TABLE]

but as it was present above that boundednes of partial sums does not hold from $H_{1}$ to $L_{1}$, However, we have inequality (3).

On the other hand, in one-dimensional, Fujji [6] and Simon [17] proved that maximal operator Fejér means is bounded from $H_{1}$ to $L_{1}$. It follows that

[TABLE]

So, natural question has arised that if inequality (4) holds true, which would be generalization of inequality (5) or we have negative answer on this problem.

In this paper we prove that there exists a function $f\in H_{1}$ such that

[TABLE]

This paper is organized as follows: in order not to disturb our discussions later on some definitions and notations are presented in Section 2. For the proofs of the main results we need some auxiliary Lemmas. These results are presented in Section 3. The formulation and detailed proof of main results can be found in Section 4.

2. Definitions and Notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers, $\mathbb{N}:=\mathbb{N}_{+}\cup\{0\}.$

Let $m:=(m_{0,}m_{1},\dots)$ denote a sequence of the positive integers not less than 2.

Denote by

[TABLE]

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group $Z_{m_{j}}$ with the product of the discrete topologies of $Z_{m_{j}}$ *,*s.

The direct product $\mu$ of the measures

[TABLE]

is the Haar measure on $G_{m_{\text{ }}}$with $\mu\left(G_{m}\right)=1.$

If $\sup_{n\in\mathbb{N}}m_{n}<\infty$, then we call $G_{m}$ a bounded Vilenkin group. If the generating sequence $m$ is not bounded then $G_{m}$ is said to be an unbounded Vilenkin group. In this paper we discuss bounded Vilenkin groups only.

The elements of $G_{m}$ are represented by sequences

[TABLE]

It is easy to give a base for the neighbourhood of $G_{m}$

[TABLE]

[TABLE]

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and $\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$ .

Let

[TABLE]

If we define the so-called generalized number system based on $m$ in the following way:

[TABLE]

then every $n\in\mathbb{N}$ can be uniquely expressed as $n=\sum_{k=0}^{\infty}n_{j}M_{j}$ where $n_{j}\in Z_{m_{j}}$ $~{}(j\in\mathbb{N})$ and only a finite number of $n_{j}`$s differ from zero. Let $\left|n\right|:=\max$ $\{j\in\mathbb{N};$ $n_{j}\neq 0\}.$

For the natural number $n=\sum_{j=1}^{\infty}n_{j}M_{j},$ we define

[TABLE]

where $\ominus$ is the inverse operation for $a_{k}\oplus b_{k}=\left(a_{k}+b_{k}\right)$mod$m_{k}.$

We define functions $v$ and $v^{\ast}$ by

[TABLE]

Next, we introduce on $G_{m}$ an orthonormal system which is called the Vilenkin system.

At first define the complex valued function $r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher functions as

[TABLE]

Now define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$ as:

[TABLE]

Specially, we call this system the Walsh-Paley one if $m\equiv 2.$

The norm (or quasi norm) of the space $L_{p}(G_{m})$ is defined by

[TABLE]

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)\,$ (for details see e.g. [1, 26]).

If $\ f\in L_{1}\left(G_{m}\right)$ we can establish Fourier coefficients, partial sums of the Fourier series, Fejér means, Dirichlet kernels with respect to the Vilenkin system in the usual manner:

[TABLE]

Recall that

[TABLE]

and

[TABLE]

The $n$-th Lebesgue constant is defined in the following way

[TABLE]

If $f\in L_{1}(G_{m}),$ the maximal functions are also be given by

[TABLE]

Hardy martingale space $H_{1}\left(G_{m}\right)$ consist of all martingales for which (for details see e.g. [27, 28])

[TABLE]

3. Auxiliary results

Lemma 1**.**

[11]** Let $n\in\mathbb{N}$. Then

[TABLE]

where $\lambda=\sup_{n\in\mathbb{N}}m_{n}.$

Lemma 2**.**

[12]** Let $n\in\mathbb{N}$. Then there exists an ansolute constant $c,$ such that

[TABLE]

4. Main Result

Theorem 1**.**

There exists a martingale $f\in H_{1},$ such that

[TABLE]

5. Proof of the Theorem

Proof of Theorem 1.

Let $\left\{\alpha_{k}:k\in\mathbb{N}\right\}$ be an increasing sequence of the positive integers such that

[TABLE]

Let

[TABLE]

where

[TABLE]

It is evident that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

Since (see equality (6))

[TABLE]

by combining (2) and (8) we get that

[TABLE]

Moreover,

[TABLE]

Let $M_{\alpha_{k}}\leq j<M_{\alpha_{k+1}}.$ Since

[TABLE]

if we apply (9) we obtain that

[TABLE]

In view of (1) we can write that

[TABLE]

By combining Lemma 1 and (15) we get that

[TABLE]

Hence, according to Lemma 2 we can conclude that

[TABLE]

The proof is complete. ∎