Tauberian conditions for \(w\)-almost convergent double sequences (Q1041510)

This paper discusses bivariate analogues of the author's paper [Positivity 13, No. 4, 611--619 (2009; Zbl 1186.40007)]. Let \((X,\|.\|)\) denote a Banach space and let \(\Im =\{ f_{n,m},n,m\geq 0\} \) denote a double sequence in \(X\). The de la Vallée-Poussin mean and the Cesàro mean are defined as \( V_{m,n}^{M,N}(\Im )=\frac{1}{NM}\sum_{j=m}^{m+M-1}\sum_{k=n}^{n+N-1}f_{j,k}\) and \(\sigma _{m,n}(\Im )=V_{0,0}^{m+1,n+1}(\Im )\). The authors consider the following limit statements. In each case \(f\in X\). (A) \(f_{m,n}@>\|.\|>>f\) if \( \|f_{m,n}-f\|\rightarrow 0\) as \(\min (m,n)\rightarrow \infty \); (B) \(f_{m,n}@>ac>>f\) if \(\sup_{m,n\geq 0}\|V_{m,n}^{M,N}(\Im )-f\|\rightarrow 0\) as \(\min (M,N)\rightarrow \infty \); (C) \(f_{m,n}@>w-ac>>f\) if \(\sup_{m\geq M,n\geq N}\|V_{m,n}^{M,N}(\Im )-f\|\rightarrow 0\) as \(\min (M,N)\rightarrow \infty \); (D) \(f_{m,n}@>(C,1,1)>>f\) if \(\|\sigma _{m,n}(\Im )-f\|\rightarrow 0\) as \(\min (m,n)\rightarrow \infty \). The author proves that for bounded sequences, (B) is equivalent to (C) and (C) implies (D). In the paper the author proves several Tauberian theorems. Among others, he proves the following result: we have (A) if and only if (C) and \(\|f_{n,m+1}-f_{n,m}\|\rightarrow 0\), \(\|f_{n+1,m}-f_{n,m}\|\rightarrow 0\) as \(\min (n,m)\rightarrow \infty \). The author also provides several types of Tauberian conditions to go from (D) to (A). As an application the author considers the spaces \((L^{p}(T^{2}),\|.\|_{p})\) and \( (C(T^{2}),\|.\|_{\infty })\) and the sequence \( f_{m,n}=s_{m,n}(f)\), where \(s_{m,n}(f)\) denotes the partial sum of the Fourier series of \(f\).