Tauberian theorems for double sequences that are statistically summable (C, 1, 1)

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:286

Article

Móricz, F

关键词: double sequence;    convergence in Pringsheim's sense;    statistical convergence;    statistical summability (C, 1, 1);    one-sided and two-sided Tauberian conditions;    slow decrease;    condition of Landau type;    slow oscillation;    condition of Hardy type;   

DOI  :  10.1016/S0022-247X(03)00529-8

来源: Elsevier

【 摘 要 】

Let (x(jk): j, k = 0, 1, 2, . . .) be a double sequence of real or complex numbers, and set sigma(mn) := (m + 1)(-1)(n + 1)-1 Sigma(j=0)(m)Sigma(k=0)(n)x(jk) for m, n = 0, 1, 2, . . . We give necessary and sufficient conditions, under which st-lim sigma(mn) = xi implies st-lim x(jk) = xi, where xi is a finite number. These Tauberian conditions are one-sided when the x(jk) are real numbers, and they are two-sided when the x(jk) are complex numbers. In particular, these Tauberian conditions are clearly satisfied if (x(jk)) is statistically slowly decreasing in the case of real sequences or if (x(jk)) is statistically slowly oscillating in the case of complex sequences. (C) 2003 Elsevier Inc. All rights reserved.

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