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Almost convergence of double sequences and strong regularity of summability matrices. (English) Zbl 0675.40004

A double sequence $x=\left\{{x}_{jk}:$ $j,k=0,1,···\right\}$ of real numbers is called almost convergent to a limit s if

$\underset{p,q\to \infty }{lim}\underset{m,n\ge 0}{sup}|1/pq\sum _{j=m}^{m+p-1}\sum _{k=n}^{n+q-1}{x}_{jk}-s|=0$

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 ${lim}_{p,q\to \infty }{\sum }_{j=0}^{p}{\sum }_{k=0}^{q}{a}_{jk}^{mn}{x}_{jk}={y}_{mn},$ ${lim}_{m,n\to \infty }{y}_{mn}=t$, where $A=\left[{a}_{jk}^{mn}:$ $j,k=0,1,···\right]$ 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

$\underset{m,n\to \infty }{lim}\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }|{{\Delta }}_{10}{a}_{jk}^{mn}|=0,\phantom{\rule{1.em}{0ex}}\underset{m,n\to \infty }{lim}\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }|{{\Delta }}_{01}{a}_{jk}^{mn}|=0,$

where ${{\Delta }}_{10}{a}_{jk}^{mn}={a}_{jk}^{mn}-{a}_{j+1,k}^{mn}$ and ${{\Delta }}_{01}{a}_{jk}^{mn}={a}_{jk}^{mn}-{a}_{j,k+1}^{mn}$ $\left(j,k,m,n=0,1,···\right)$. Then the authors define $A=\left[{q}_{jk}^{MN}\right]$ as a hump matrix if (i) for each m, n, k there exists a positive integer $p=p\left(m,n,k\right)$ such that ${a}_{jk}^{mn}\le {a}_{j+1,k}^{mn}$ if $0\le j and ${a}_{jk}^{mn}\ge {a}_{j+1,k}^{mn}$ if $j\ge p$; (ii) for each m, n, j there exists a positive integer $q=q\left(m,n,j\right)$ such that ${a}_{jk}^{mn}\le {a}_{j,k+1}^{mn}$ if $0\le k and ${a}_{jk}^{mn}\ge {a}_{j,k+1}^{mn}$ if $k\ge q$. Let $ℋ$ be the set of all hump matrices $A=\left[{a}_{jk}^{mn}\right]$ which are bounded regular and for which ${lim}_{m,n\to \infty }{\sum }_{j=0}^{\infty }{sup}_{k\ge 0}|{a}_{jk}^{mn}|=0$ and ${lim}_{m,n\to \infty }{\sum }_{k=0}^{\infty }{sup}_{j\ge 0}|{a}_{jk}^{mn}|=0·$ Let $𝔞c$ be the set of all double sequences x which are almost convergent and ${C}_{A}$ be the set of all bounded double sequences whose A-means converge, then it is shown that $𝔞c={\cap }_{A\in ℋ}{C}_{A}$.

Reviewer: D.P.Gupta

##### MSC:
 40C05 Matrix methods in summability 42B05 Fourier series and coefficients, several variables
##### Keywords:
strongly regular matrix; hump matrix