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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 sup m,n0 |1/pq j=m m+p-1 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 j=0 p k=0 q a jk mn x jk =y mn , lim m,n y mn =t, where A=[a jk mn : 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 j=0 k=0 |Δ 10 a jk mn |=0,lim m,n j=0 k=0 |Δ 01 a jk mn |=0,

where Δ 10 a jk mn =a jk mn -a j+1,k mn and Δ 01 a jk mn =a jk mn -a j,k+1 mn (j,k,m,n=0,1,···). Then the authors define A=[q jk MN ] as a hump matrix if (i) for each m, n, k there exists a positive integer p=p(m,n,k) such that a jk mn a j+1,k mn if 0j<p and a jk mn a j+1,k mn if jp; (ii) for each m, n, j there exists a positive integer q=q(m,n,j) such that a jk mn a j,k+1 mn if 0k<q and a jk mn a j,k+1 mn if kq. Let be the set of all hump matrices A=[a jk mn ] which are bounded regular and for which lim m,n j=0 sup k0 |a jk mn |=0 and lim m,n k=0 sup j0 |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= A C A .

Reviewer: D.P.Gupta

MSC:
40C05Matrix methods in summability
42B05Fourier series and coefficients, several variables