On absolute matrix summability of orthogonal series

Abstract

In this paper, we prove two theorems on $|A\text{,}\delta {|}_k\text{,}1⩽k⩽2\text{,}\delta ⩾0$, summability of orthogonal series. Also, several known and new results are deduced as corollaries of the main results.

Introduction

Let ${\sum }_{n=0}^{\infty }{a}_n$ be a given infinite series with its partial sums $s_n\text{,}(C\text{,}\alpha )$ the Cesàro matrix of order $\alpha$. If ${\sigma }_n^{\alpha }$ denotes the nth term of the $(C\text{,}\alpha )$-transformation of $s≔\{s_n\}$, then Flett [2] defined absolute summability of order $k⩾1$ as follows. A series ${\sum }_{n=0}^{\infty }{a}_n$ is said to be summable $|C\text{,}\alpha {|}_k\text{,}k⩾1$ if

$${\displaystyle \sum _{n=1}^{\infty }}{n}^{k-1}|{\sigma }_n^{\alpha }-{\sigma }_{n-1}^{\alpha }{|}^k<\infty \text{.}$$

In an effort to extend (1.1) to other classes of matrices, some authors have interpreted the n in (1.1) to represent the reciprocal of the nth diagonal entry of the matrix.

For example, in [1] with $t_n$ denoting the nth term of the weighted mean transform of a sequence $\{s_n\}$, i.e.,

$$t_n=\frac{1}{P_n}{\displaystyle \sum _{v=0}^n}{p}_v{s}_v\text{,}$$

their version of (1.1) becomes

$${\displaystyle \sum _{n=1}^{\infty }}{\left(\frac{P_n}{p_n}\right)}^{k-1}|t_n-t_{n-1}{|}^k<\infty \text{.}$$

Detailed arguments showing that (1.2) is not an appropriate extension of (1.1) appear in [7], and so will not be repeated here.

The following definitions can be reduced to the correct definitions of such extensions.

Let $A≔(a_{\mathit{nv}})$ be a normal matrix, i.e. a lower triangular matrix of non-zero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence $s≔\{s_n\}$ to $\mathit{As}≔\{A_n(s)\}$, where

$$A_n(s)≔{\displaystyle \sum _{v=0}^n}{a}_{\mathit{nv}}{s}_v\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$

The infinite series ${\sum }_{n=0}^{\infty }{a}_n$ is said to be absolutely summable $|A{|}_k$ ($|A\text{,}\delta {|}_k$), $k⩾1\text{,}\delta ⩾0$, if (see [2])

$${\displaystyle \sum _{n=1}^{\infty }}{n}^{k-1}|\overline{\mathrm{\Delta }}{A}_n(s){|}^k\left({\displaystyle \sum _{n=1}^{\infty }}{n}^{\delta k+k-1}|\overline{\mathrm{\Delta }}{A}_n(s){|}^k\right)$$

converges, where

$$\overline{\mathrm{\Delta }}{A}_n(s)=A_n(s)-A_{n-1}(s)$$

and we write in brief

$${\displaystyle \sum _{n=0}^{\infty }}{a}_n\in |A{|}_k\left(\in |A\text{,}\delta {|}_k\right)\text{,}$$

respectively.

Let $\{{\phi }_n(x)\}$ be an orthonormal system defined in the interval $(a\text{,}b)$. We assume that $f(x)$ belongs to $L^2(a\text{,}b)$ and

$$f(x)\sim {\displaystyle \sum _{n=0}^{\infty }}{c}_n{\phi }_n(x)\text{,}$$

where $c_n={\int }_a^{b}f(x){\phi }_n(x)\mathit{dx}\text{,}(n=0\text{,}1\text{,}2\text{,}\dots )$.

Among others, Y. Okuyama [3] concerning the $|N\text{,}{p}_n{|}_k\text{,}1⩽k⩽2$, summability of orthogonal series (1.3) proved the following two theorems. The first one is

Theorem 1.1

Let $1⩽k⩽2$ and $\{{\lambda }_n\}$ be a positive sequence. If $\{p_n\}$ is a positive sequence and the series

$${\displaystyle \sum _{n=0}^{\infty }}\frac{p_n}{P_n{P}_{n-1}^k}{\left\{{\displaystyle \sum _{j=1}^n}{p}_{n-j}^2{\left(\frac{P_n}{p_n}-\frac{P_{n-j}}{p_{n-j}}\right)}^2{\lambda }_j^2|c_j{|}^2\right\}}^{\frac{k}{2}}$$

converges, then the orthogonal series

$${\displaystyle \sum _{n=0}^{\infty }}{\lambda }_n{c}_n{\phi }_n(x)$$

is summable $|N\text{,}{p}_n{|}_k$ almost everywhere.

For the second one, first of all he wrote

$$w^{(k)}(j)=j^{-1}{\displaystyle \sum _{n=j}^{\infty }}\frac{n^{\frac{2}{k}}{p}_n^{\frac{2}{k}}{p}_{n-j}^2}{P_n^{2+\frac{2}{k}}}{\left(\frac{P_n}{p_n}-\frac{P_{n-j}}{p_{n-j}}\right)}^2\text{.}$$

Theorem 1.2

Let $1⩽k⩽2$ and $\{\mathrm{\Omega }(n)\}$ be a positive sequence such that $\{\mathrm{\Omega }(n)/n\}$ is a non-increasing sequence and the series ${\sum }_{n=1}^{\infty }\frac{1}{n\mathrm{\Omega }(n)}$ converges. If $\{p_n\}$ is a positive non-increasing sequence and the series ${\sum }_{n=1}^{\infty }|c_n{|}^2{\mathrm{\Omega }}^{\frac{2}{k}-1}(n)w^{(k)}(n)$ converges, then the orthogonal series ${\sum }_{n=0}^{\infty }{c}_n{\phi }_n(x)$ is $|N\text{,}{p}_n{|}_k$ summable almost everywhere.

Theorem 1.1 includes a result of Singh [8] which is an extension, for trigonometric series, of theorems due to Pati [6], Ul’yanov [9] and Wang [10], until Theorem 1.2 generalize a theorem of Okuyama [4].

The main purpose of the present paper is to generalize Theorem 1.1, Theorem 1.2 for $|A\text{,}\delta {|}_k$ summability of the orthogonal series (1.3), where $1⩽k⩽2\text{,}\delta ⩾0$. Before starting the main results first introduce some further notations.

Given a normal matrix $A≔(a_{\mathit{nv}})$, we associate two lower semi matrices $\bar{A}≔({\overline{a}}_{\mathit{nv}})$ and $\widehat{A}≔({\hat{a}}_{\mathit{nv}})$ as follows:

$${\overline{a}}_{\mathit{nv}}≔{\displaystyle \sum _{i=v}^n}{a}_{\mathit{ni}}\text{,}n\text{,}i=0\text{,}1\text{,}2\text{,}\dots$$

and

$${\hat{a}}_{00}={\overline{a}}_{00}=a_{00}\text{,}{\hat{a}}_{\mathit{nv}}={\overline{a}}_{\mathit{nv}}-{\overline{a}}_{n-1\text{,}v}\text{,}n=1\text{,}2\text{,}\dots$$

It may be noted that $\bar{A}$ and $\hat{A}$ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively.

Throughout this paper K denotes a positive constant that it may depends only on k, and be different in different relations.