Tauberian theorems for Cesàro summable double sequences

• Autorzy: Ferenc Móricz
• Języki publikacji: EN

Abstrakty

EN

$(s_{jk}: j,k = 0,1,...)$ 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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 40E05: Tauberian theorems, general

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 110
• Numer: 1
• Strony: 83-96

Daty

wydano

1994

otrzymano

1993-08-31

Twórcy

autor

Ferenc Móricz

• Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary

Bibliografia

• [1] G. H. Hardy, Divergent Series, Univ. Press, Oxford, 1956.
• [2] E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy und Axer, Prace Mat.-Fiz. 21 (1910), 97-177.
• [3] I. J. Maddox, A Tauberian theorem for ordered spaces, Analysis 9 (1989), 297-302.
• [4] F. Móricz, Necessary and sufficient Tauberian conditions, under which convergence follows from summability (C,1), Bull. London Math. Soc., to appear.
• [5] R. Schmidt, Über divergente Folgen und lineare Mittelbindungen, Math. Z. 22 (1925), 89-152.
• [6] A. Zygmund, Trigonometric Series, Univ. Press, Cambridge, 1959.