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Zbl 0675.40004
Almost convergence of double sequences and strong regularity of summability matrices.
(English)
[J] Math. Proc. Camb. Philos. Soc. 104, No.2, 283-294 (1988). ISSN 0305-0041; ISSN 1469-8064/e

A double sequence $x=\{x\sb{jk}:$ $j,k=0,1,...\}$ of real numbers is called almost convergent to a limit s if $$\lim\sb{p,q\to \infty}\sup\sb{m,n\ge 0}\vert 1/pq\sum\sp{m+p-1}\sb{j=m}\sum\sp{n+q- 1}\sb{k=n}x\sb{jk}-s\vert =0$$ uniformly in m and n. The definition is an extension of Lorentz's definition of almost convergence of single sequences. The sequence x is said to be A-summable to limit t if $\lim\sb{p,q\to \infty}\sum\sp{p}\sb{j=0}\sum\sp{q}\sb{k=0}a\sp{mn}\sb{jk} x\sb{jk}=y\sb{mn},$ $\lim\sb{m,n\to \infty}y\sb{mn}=t$, where $A=[a\sp{mn}\sb{jk}:$ $j,k=0,1,...]$ is doubly infinite matrix of real numbers for all $m,n=0,1,...$. The matrix A is said to be bounded- regular if every bounded and convergent sequence x is A-summable to the same limit and the A-means are also bounded. The matrix A is strongly regular if every almost convergent sequence x is A-summable to the same limit and the A-means are also bounded. It is shown that the necessary and sufficient conditions for a matrix A to be strongly regular are that A is bounded-regular and $$\lim\sb{m,n\to \infty}\sum\sp{\infty}\sb{j=0}\sum\sp{\infty}\sb{k=0}\vert \Delta\sb{10}a\sp{mn}\sb{jk}\vert =0,\quad \lim\sb{m,n\to \infty}\sum\sp{\infty}\sb{j=0}\sum\sp{\infty}\sb{k=0}\vert \Delta\sb{01}a\sp{mn}\sb{jk}\vert =0,$$ where $\Delta\sb{10}a\sp{mn}\sb{jk}=a\sp{mn}\sb{jk}-a\sp{mn}\sb{j+1,k}$ and $\Delta\sb{01}a\sp{mn}\sb{jk}=a\sp{mn}\sb{jk}-a\sp{mn}\sb{j,k+1}$ $(j,k,m,n=0,1,...)$. Then the authors define $A=[q\sp{MN}\sb{jk}]$ as a hump matrix if (i) for each m, n, k there exists a positive integer $p=p(m,n,k)$ such that $a\sp{mn}\sb{jk}\le a\sp{mn}\sb{j+1,k}$ if $0\le j<p$ and $a\sp{mn}\sb{jk}\ge a\sp{mn}\sb{j+1,k}$ if $j\ge p$; (ii) for each m, n, j there exists a positive integer $q=q(m,n,j)$ such that $a\sp{mn}\sb{jk}\le a\sp{mn}\sb{j,k+1}$ if $0\le k<q$ and $a\sp{mn}\sb{jk}\ge a\sp{mn}\sb{j,k+1}$ if $k\ge q$. Let ${\cal H}$ be the set of all hump matrices $A=[a\sp{mn}\sb{jk}]$ which are bounded regular and for which $\lim\sb{m,n\to \infty}\sum\sp{\infty}\sb{j=0}\sup\sb{k\ge 0}\vert a\sp{mn}\sb{jk}\vert =0$ and $\lim\sb{m,n\to \infty}\sum\sp{\infty}\sb{k=0}\sup\sb{j\ge 0}\vert a\sp{mn}\sb{jk}\vert =0.$ Let ${\frak ac}$ be the set of all double sequences x which are almost convergent and $C\sb A$ be the set of all bounded double sequences whose A-means converge, then it is shown that ${\frak ac}=\cap\sb{A\in {\cal H}}C\sb A$.
[D.P.Gupta]
MSC 2000:
*40C05 Matrix methods in summability
42B05 Fourier series and coefficients, several variables

Keywords: strongly regular matrix; hump matrix

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