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


40C05 Matrix methods for summability
42B05 Fourier series and coefficients in several variables
Full Text: DOI


[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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