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

A double sequence $$x=\{x_{jk}:$$ $$j,k=0,1,...\}$$ of real numbers is called almost convergent to a limit s if $\lim_{p,q\to \infty}\sup_{m,n\geq 0}| 1/pq\sum^{m+p-1}_{j=m}\sum^{n+q- 1}_{k=n}x_{jk}-s| =0$ 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 $$\lim_{p,q\to \infty}\sum^{p}_{j=0}\sum^{q}_{k=0}a^{mn}_{jk} x_{jk}=y_{mn},$$ $$\lim_{m,n\to \infty}y_{mn}=t$$, where $$A=[a^{mn}_{jk}:$$ $$j,k=0,1,...]$$ 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 $\lim_{m,n\to \infty}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0}| \Delta_{10}a^{mn}_{jk}| =0,\quad \lim_{m,n\to \infty}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0}| \Delta_{01}a^{mn}_{jk}| =0,$ where $$\Delta_{10}a^{mn}_{jk}=a^{mn}_{jk}-a^{mn}_{j+1,k}$$ and $$\Delta_{01}a^{mn}_{jk}=a^{mn}_{jk}-a^{mn}_{j,k+1}$$ $$(j,k,m,n=0,1,...)$$. Then the authors define $$A=[q^{MN}_{jk}]$$ as a hump matrix if (i) for each m, n, k there exists a positive integer $$p=p(m,n,k)$$ such that $$a^{mn}_{jk}\leq a^{mn}_{j+1,k}$$ if $$0\leq j<p$$ and $$a^{mn}_{jk}\geq a^{mn}_{j+1,k}$$ if $$j\geq p$$; (ii) for each m, n, j there exists a positive integer $$q=q(m,n,j)$$ such that $$a^{mn}_{jk}\leq a^{mn}_{j,k+1}$$ if $$0\leq k<q$$ and $$a^{mn}_{jk}\geq a^{mn}_{j,k+1}$$ if $$k\geq q$$. Let $${\mathcal H}$$ be the set of all hump matrices $$A=[a^{mn}_{jk}]$$ which are bounded regular and for which $$\lim_{m,n\to \infty}\sum^{\infty}_{j=0}\sup_{k\geq 0}| a^{mn}_{jk}| =0$$ and $$\lim_{m,n\to \infty}\sum^{\infty}_{k=0}\sup_{j\geq 0}| a^{mn}_{jk}| =0.$$ Let $${\mathfrak ac}$$ 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 $${\mathfrak ac}=\cap_{A\in {\mathcal H}}C_ A$$.
Reviewer: D.P.Gupta

### MSC:

 40C05 Matrix methods for summability 42B05 Fourier series and coefficients in several variables

### Keywords:

strongly regular matrix; hump matrix
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### References:

 [1] DOI: 10.2307/1989172 · JFM 52.0223.01 [2] DOI: 10.1007/BF02393648 · Zbl 0031.29501 [3] Rhoades, Approximation Theory III pp 735– (1980) [4] Rhoades, Approximation Theory and Applications pp 173– (1985)
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