Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz

Studia Mathematica, Tome 110 (1994) no. 1, pp. 83-96

Résumé

$(s_{jk}: j,k = 0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_{jk})$ converges in Pringsheim's sense. These conditions are satisfied if $(s_{jk})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_{jk})$ is summable (C,1,1) to a finite limit and there exist constants $n_1 > 0$ and H such that $jk(s_{jk} - s_{j-1,k} - s_{j-1,k} + s_{j-1,k-1}) ≥ -H$, $j(s_{jk} - s_{j-1, k}) ≥ -H$ and $k(s_{jk} - s_{j,k-1}) ≥ -H$ whenever $j,k > n_1$, then $(s_{jk})$ converges. We always mean convergence in Pringsheim's sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.

Détail

DOI : 10.4064/sm-110-1-83-96

Keywords: double sequence, convergence in Pringsheim's sense, summability (C, 1, 1), (C, 1, 0) and (C, 0, 1), one-sided Tauberian condition of Landau and Hardy type, slow decrease, ordered linear space