Some inequalities for Ces\`aro Means of double Vilenkin-Fourier Series

TL;DR

This paper establishes new inequalities that describe how well Cesàro means approximate functions in the $L^p$ space using double Vilenkin-Fourier series, enhancing understanding of convergence rates.

Contribution

It introduces novel inequalities that relate to the approximation rate of Cesàro means for double Vilenkin-Fourier series in $L^p$ spaces, advancing theoretical knowledge.

Findings

New inequalities for Cesàro means approximation rates

Improved understanding of convergence in $L^p$ spaces

Enhanced bounds for double Vilenkin-Fourier series

Abstract

In this paper we state and prove some new inequalities related to the rate of $L^{p}$ approximation by Ces\`aro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from $L^{p}$.

Equations

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

$$x+y=\left(\left(x_0+y_0\right)mod{m}_0,...,\left(x_k+y_k\right)mod{m}_k,...\right),$$

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

$$I_n\left(x\right):=\{y\in G_m|y_0=x_0,...,y_{n-1}=x_{n-1}\}$$

$$r_k(x):=\exp \frac{2\pi i{x}_k}{m_k}\left(i^2=-1,x\in G_m,k\in N\right).$$

$${\psi }_n\left(x\right):=\prod _{k=0}^{\infty }{r}_k^{n_k}\left(x\right),\left(nϵN\right).$$

$$D_n:=\sum _{k=0}^{n-1}{\psi }_k,\left(n\in N_{+}\right).$$

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

$${\Vert f\Vert }_p:={\left(\underset{G_m^2}{\int }{\left|f\left(x,y\right)\right|}^{p}d\mu \left(x,y\right)\right)}^{1/p},\left(1\le p<\infty \right).$$

$$\hat{f}\left(n_1,n_2\right):=\underset{G_m^2}{\int }f\left(x,y\right){\bar{\psi }}_{n_1}\left(x\right){\bar{\psi }}_{n_2}\left(y\right)d\mu \left(x,y\right),$$

$$S_{n_{1},n_{2}}\left(x,y,f\right):=\sum\limits_{k_{1}=0}^{n_{1}-1}\sum\limits_{k_{2}=0}^{n_{2}-1}\widehat{f}\left(k_{1},k_{2}\right)\psi_{k_{1}}\left(x\right)\psi_{k_{2}}\left(y\right),\$$

$$D_{n_1,n_2}\left(x,y\right):=D_{n_1}\left(x\right)D_{n_2}\left(y\right),$$

$$S_{n}^{\left(1\right)}\left(x,y,f\right):=\sum\limits_{l=0}^{n-1}\widehat{f}\left(l,y\right)\bar{\psi}_{l}\left(x\right),\$$

$$S_{m}^{\left(2\right)}\left(x,y,f\right):=\sum\limits_{r=0}^{m-1}\widehat{f}\left(x,r\right)\bar{\psi}_{r}\left(y\right),\$$

$$\hat{f}\left(l,y\right)=\underset{G_m}{\int }f\left(x,y\right){\psi }_l\left(x\right)d\mu \left(x\right)$$

$$\hat{f}\left(x,r\right)=\underset{G_m}{\int }f\left(x,y\right){\psi }_r\left(y\right)d\mu \left(y\right).$$

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

$$A_0^{\alpha }=1,A_n^{\alpha }=\frac{\left(\alpha +1\right)...\left(\alpha +n\right)}{n!}.$$

$$A_n^{\alpha }=\sum _{k=0}^n{A}_k^{\alpha -1}.$$

$$A_{n}^{\alpha}-A_{n-1}^{\alpha}=A_{n}^{\alpha-1}.\ \ \ \ \ \ \$$

$$c_{1}(\alpha)n^{\alpha}\leq A_{n}^{\alpha}\leq c_{2}(\alpha)n^{\alpha},\ \ \ \ \ \ \$$

$${\omega }_1{\left(f,\frac{1}{M_n}\right)}_p=\underset{u\in I_n}{\sup }{\Vert f\left(\cdot +u,\cdot \right)-f\left(\cdot ,\cdot \right)\Vert }_p,$$

$${\omega }_2{\left(f,\frac{1}{M_n}\right)}_p=\underset{v\in I_n}{\sup }{\Vert f\left(\cdot ,\cdot +v\right)-f\left(\cdot ,\cdot \right)\Vert }_p,$$

$${\omega }_{1,2}{\left(f,\frac{1}{M_n},\frac{1}{M_m}\right)}_p$$

$$=\underset{\left(u,v\right)\in I_n\times I_m}{\sup }{\Vert f\left(\cdot +u,\cdot +v\right)-f\left(\cdot +u,\cdot \right)-f\left(\cdot ,\cdot +v\right)+f\left(\cdot ,\cdot \right)\Vert }_p.$$

$${\omega }_{1,2}{\left(f,\frac{1}{M_n},\frac{1}{M_m}\right)}_p\le {\omega }_1{\left(f,\frac{1}{M_n}\right)}_p+{\omega }_2{\left(f,\frac{1}{M_m}\right)}_p.$$

$$\omega {\left(f,\frac{1}{M_n}\right)}_p=\underset{\left(u,v\right)\in I_n\times I_n}{\sup }{\Vert f\left(\cdot +u,\cdot +v\right)-f\left(\cdot ,\cdot \right)\Vert }_p.$$

$${\Vert {\sigma }_{2^k}^{-\alpha }\left(f\right)-f\Vert }_p\le c\left(\alpha \right)\{2^{k\alpha }{\omega }_1{\left(f,1/2^{k-1}\right)}_p+2^{k\alpha }{\omega }_2{\left(f,1/2^{k-1}\right)}_p+$$

$$+\sum _{r=0}^{k-2}{2}^{r-k}{\omega }_1{\left(f,1/2^r\right)}_p+\sum _{s=0}^{k-2}{2}^{s-k}{\omega }_2{\left(f,1/2^s\right)}_p\}$$

$${\Vert {\sigma }_n^{-\alpha }\left(f\right)-f\Vert }_p\le c\left(\alpha \right)\{2^{k\alpha }k{\omega }_1{\left(f,1/2^{k-1}\right)}_p+2^{k\alpha }k{\omega }_2{\left(f,1/2^{k-1}\right)}_p+$$

Full text

1 ; The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.

2The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.

2000 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Inequalities, Approximation, Vilenkin system, Vilenkin-Fourier series, Cesàro means, Convergence in norm.

1.

Introduction

Let $N_{+}$ denote the set of positive integers, $N:=N_{+}\cup\{0\}.$ Let $m:=\left(m_{0},m_{1},...\right)$ denote a sequence of positive integers not less then 2. Denote by $Z_{m_{k}}:=\{0,1,...,m_{k}-1\}$ the additive group of integers modulo $m_{k}$. Define the group $G_{m}$ as the complete direct product of the groups $Z_{m_{j}}$, with the product of the discrete topologies of $Z_{mj}$’s.

The direct product of the measures

[TABLE]

is the Haar measure on $G_{m}$ with $\mu\left(G_{m}\right)=1.$ If the sequence $m$ is bounded, then $G_{m}$ is called a bounded Vilenkin group. In this paper we will consider only bounded Vilenkin groups. The elements of $G_{m}$ can be represented by sequences $x:=\left(x_{0},x_{1},...,x_{j},...\right),$ $\left(x_{j}\in Z_{m_{j}}\right).$ The group operation $+$ in $G_{m}$ is given by

[TABLE]

where $x:=\left(x_{0},...,x_{k},...\right)$ and $y:=\left(y_{0},...,y_{k},...\right)\in G_{m}.$ The inverse of $+$ will be denoted by $-.$

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

[TABLE]

[TABLE]

for $x$ $\in$ $G_{m},$ $n$ $\in$ $N.$ Define $I_{n}:=I_{n}\left(0\right)$ for $n\in N_{+}$. Set $e_{n}:=\left(0,...,0,1,0,...\right)\in$ $G_{m}$ the $n$ th coordinate of which is $1$ and the rest are zeros $\left(n\in N\right).$

If we define the so-called generalized number system based on $m$ in the following way: $M_{0}:=1,$ $M_{k+1}:=m_{k}M_{k}$ $\left(k\in N\right),$ then every $n$ $\in$ $N$ can be uniquely expressed as $n=\sum\limits_{j=0}^{\infty}n_{j}M_{j},$ where $n_{j}$ $\in\ Z_{m_{j}}$ $\left(j\in N_{+}\right)$ and only a finite number of $n_{j}$’s differ from zero. We also use the following notation: $\left|n\right|:=$max$\{k\in N:n_{k}\neq 0\}$ (that is , $M_{|n|}\leq n<M_{|n|+1}$, $n\neq 0$). For every $x\in G_{m}$ we denote $|x|:=\sum\limits_{j=0}^{\infty}\frac{x_{j}}{M_{j+1}},\left(x_{j}\text{ }\in Z_{m_{j}}\right)$.

Next, we introduce on $G_{m}$ an orthonormal system, which is called Vilenkin system. At first define the complex valued functions $r_{k}\left(x\right):G_{m}\rightarrow C$, the generalized Rademacher functions, in this way:

[TABLE]

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

[TABLE]

In particular, we call the system the Walsh-Paley system if $m=2.$ Each $\psi_{n}$ is a character of $G_{m}$ and all characters of $G_{m}$ are of this norm. Moreover, $\psi_{n}\left(-x\right)=\bar{\psi}_{n}\left(x\right)$.

The Dirichlet kernels are defined by

[TABLE]

Recall that (see [4] or [25])

[TABLE]

The Vilenkin system is orthonormal and complete in  $L^{1}\left(G_{m}\right)$ ( see [1]).

Next, we introduce some notation with respect to the theory of two-demonsional Vilenkin system. Let $\tilde{m}$ be a sequence like $m$. The relation between the sequences $\left(\tilde{m}_{n}\right)$ and  $\left(\tilde{M}_{n}\right)$ is the same as between sequences $\left(m_{n}\right)$ and $\left(M_{n}\right).$ The group $G_{m}\times G_{\tilde{m}}$ is called a two-dimensional Vilenkin group. The normalized Haar measure is denoted by $\mu$ as in the one-dimensional case. We also suppose that $m=\tilde{m}$ and $G_{m}\times G_{\tilde{m}}=G_{m}^{2}.$

The norm of the space $L^{p}\left(G_{m}^{2}\right)$ is defined by

[TABLE]

Denote by $C\left(G_{m}^{2}\right)$ the class of continuous functions on the group $G_{m}^{2}$, endoved with the supremum norm.

For the sake of brevity in notation, we agree to write $L^{\infty}\left(G_{m}^{2}\right)$ instead of $C\left(G_{m}^{2}\right).$

The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series, the Dirichlet kernels with respect to the two-dimensional Vilenkin system are defined as follows:

[TABLE]

[TABLE]

[TABLE]

Denote

[TABLE]

[TABLE]

where

[TABLE]

and

[TABLE]

The $(C,-\alpha)$ means of double Vilenkin-Fourier series are defined as follows

[TABLE]

where

[TABLE]

It is well known that (see [29])

[TABLE]

[TABLE]

and

[TABLE]

where positive constants $c_{1}$ and $c_{2}$ are dependent on $\alpha$.

The dyadic partial moduli of continuity of a function f\in L^{p}\left(G_{m}^{2}\right)\ in the $L^{p}$-norm are defined by

[TABLE]

and

[TABLE]

while the dyadic mixed modulus of continuity is defined as follows:

[TABLE]

[TABLE]

It is clear that

[TABLE]

The dyadic total modulus of continuity is defined by

[TABLE]

The problems of summability of partial sums and Cesàro means for Walsh-Fourier series were studied in [2], [15]-[24], [27].

The convergence issue of Fejér (and Cesàro ) means on the Walsh and Vilenkin groups for unbouded case were studies in [5]-[11] .

In his monography [28] L.V. Zhizhinashvili investigated the behavior of Cesàro $(C,\alpha)-$means for double trigonometric Fourier series in detail. U.Goginava [20] studied the analogical question in case of the Walsh system. In particular, the following theorems were proved:

Theorem A**.**

Let $f$ belong to $L^{p}\left(G_{2}\right)$ for some $p$ $\in$ $\left[1,\infty\right]$ and  $\alpha$ $\in$ $\left(0,1\right)$. Then, for any  $2^{k}\leq n<2^{k+1},$ $(k,n\in N)$ , the inequality

[TABLE]

[TABLE]

holds.

Theorem B**.**

Let $f$ belong to $L^{p}\left(G_{2}\right)$ for some $p$ $\in$ $\left[1,\infty\right]$ and  $\alpha$ $\in$ $\left(0,1\right)$. Then, for any  $2^{k}\leq n<2^{k+1},$ $(k,n\in N)$ , the inequality

[TABLE]

[TABLE]

holds.

In this paper, we state and prove the analogous results in the case of double Vilenkin-Fourier series. Our main results read:

Theorem 1**.**

Let $f$ belong to $L^{p}\left(G_{m}^{2}\right)$ for some $p$ $\in$ $\left[1,\infty\right]$ and  $\alpha$ $\in$ $\left(0,1\right)$. Then, for any  $M_{k}\leq n<M_{k+1}$ $(k,n\in N)$ , the inequality

[TABLE]

[TABLE]

holds.

Theorem 2**.**

Let $f$ belong to $L^{p}\left(G_{m}^{2}\right)$ for some $p$ $\in$ $\left[1,\infty\right]$ and  $\alpha$ $\in$ $\left(0,1\right)$. Then, for any  $M_{k}\leq n<M_{k+1}$ $(k,n\in N)$ , the inequality

[TABLE]

[TABLE]

[TABLE]

holds.

In order to make the proofs of these Theorems more clear we formulate some auxiliary Lemmas in Section 2. Some of these Lemmas are new and of independent interest. The detailed proofs can be found in Section 3.

2. AUXILIARY LEMMAS

In order to prove Theorem 1 and Theorem 2 we need the following Lemmas (see [1], [3] and [12], respectively)

Lemma 1**.**

Let $\alpha_{1},\alpha_{2},...,\alpha_{n}$ be real numbers.Then

[TABLE]

Lemma 2**.**

Let $\alpha_{1},\alpha_{2},...,\alpha_{n}$ be real numbers. Then

[TABLE]

Lemma 3**.**

Let $0\leq j<n_{s}M_{s}$ and $0\leq n_{s}<m_{s}.$ Then

[TABLE]

We also need the following new Lemmas of independent interest.

Lemma 4**.**

Let $f$ belong to $L^{p}\left(G_{m}^{2}\right)$ for some $p$ $\in$ $\left[1,\infty\right]$. Then, for every  $\alpha$ $\in$ $\left(0,1\right)$, the following inequality holds

[TABLE]

[TABLE]

where $M_{k}\leq n<M_{k+1}.$

Lemma 5**.**

Let $\alpha$ $\in$ $\left(0,1\right)$ and $\ p=M_{k},M_{k}+1,....$ Then

[TABLE]

Lemma 6**.**

The inequality

[TABLE]

holds.

3. The detailed proofs

Proof of Lemma 3..

Applying Abel’s transformation, from (2) we get that

[TABLE]

[TABLE]

[TABLE]

where the first and the second terms on the right side of inequality (5) should be denoted by $I_{1}$ and $I_{2}$ respectively.

For $I_{2}$ we can estimate as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second and the third terms on the right side of inequality (6) should be denoted by $I_{21}$, $I_{22}$ and $I_{23}$ respectively.

It is evident that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence,

[TABLE]

Moreover, according to the generalized Minkowski inequality, Lemma 2 and by (1) and (4) we obtain that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The estimation of $I_{23}$ is analogous to the estimation of $I_{21}$ and we get that

[TABLE]

Analogously, we can estimate $I_{1}$ in the following way

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By combining (7)-(9) with (10) for $I$ we find that

[TABLE]

The proof of Lemma 3 is complete. ∎

Proof of Lemma 4..

It is evident that

[TABLE]

[TABLE]

where the first and the second terms on the right side of inequality (12) should be denoted by $II_{1}$ and $II_{2}$ respectively.

From (1) by $\left|A_{p-M_{k}}^{-\alpha-1}\right|\leq 1$ we get that

[TABLE]

Moreover, by Lemma 3 we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second, the third and the fourth terms on the right side of inequality (14) should be denoted by $II_{11}$, $II_{12}$, $II_{13}$ and $II_{14}$ respectively.

From (1) and (4) it follows that

[TABLE]

By Applying Abel’s transformation, in view of Lemma 2 we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The estimation of $II_{12}$ and $II_{13}$ are analogous to the estimation of $II_{11}$. By Applying Abel’s transformation, in view of Lemma 1 we find that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The proof is complete by combining (12)-(18). ∎

Proof of Lemma 5..

Let

[TABLE]

Denote

[TABLE]

Since ( see [4])

[TABLE]

we find that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second, the third, the fourth and the fifth terms on the right side of inequality (20) should be denoted by $III_{1}$, $III_{2}$, $III_{3}$, $III_{4}$ and $III_{5}$ respectively.

By (1) we have that

[TABLE]

Moreover, since (see [26] )

[TABLE]

for $III_{4}$ we get that

[TABLE]

[TABLE]

Analogously, we find that

[TABLE]

[TABLE]

For $r\in\{0,...m_{A}-1\},$ $0\leq\ j<M_{A}$ , ( see [4]), it yields that

[TABLE]

Thus, we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

On the other hand, by (1) and (4) we obtain that

[TABLE]

Consequently, for $III_{1}$ we have the estimate

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second, the third and the fourth terms on the right side of inequality (25) should be denoted by $III_{11}$, $III_{12}$, $III_{13}$ and $III_{14}$ respectively.

From Lemma 4 we have that

[TABLE]

The estimation of $III_{11}$ is analogous to the estimation of $III_{3}$ and we find that

[TABLE]

The estimation of $III_{12}$ and $III_{13}$ is analogous to the estimation of $III_{4}$ and we obtain that

[TABLE]

and

[TABLE]

After substituting (21) and (23)- (29) into (20) we conclude that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The proof is complete. ∎

Now we are ready to prove the main results

Proof of Theorem 1..

It is evident that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From Lemma 5 it follows that

[TABLE]

Moreover, for II we have the estimate

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first and the second terms on the right side of inequality (32) should be denoted by $II_{1}$ and $II_{2}$ respectively.

In view of generalized Minkowski inequality, by (4) and using Lemma 5 we get that

[TABLE]

[TABLE]

[TABLE]

The estimation of $II_{2}$ is analogous to the estimation of $II_{1}$ and we find that

[TABLE]

Combining (30)- (34) we receive the proof of Theorem 1. ∎

Proof of Theorem 2..

It is evident that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second and the third terms on the right side of inequality (35) should be denoted by $I$, $II$ and $III$ respectively.

From Lemma 4 it follows that

[TABLE]

Next, we repeat the arguments just in the same way as in the proof of Theorem 1 and find that

[TABLE]

On the other hand, for III we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first, the second and the third terms on the right side of inequality (38) should be denoted by $III_{1}$, $III_{2}$ and $III_{3}$ respectively.

It is easy to show that

[TABLE]

According to the generalized Minkowski inequality and by using Lemma 5 for $III_{1}$ we obtain that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The estimation of $III_{3}$ is analogous to the estimation of $III_{2}$ and we find that

[TABLE]

After substituting (36)- (37), (41) into (35), we receive the proof of Theorem 2. ∎

Author details

1 The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.

2 The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.

Authors’ contributions

The authors contributed equally to the writing of this paper. Both authors approved the final version of the manuscript.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The authors would like to thank the referees for helpful suggestions.