Some Tauberian theorems for the weighted mean method of summability of double sequence

Abstract

UDC 517.5 Let p = ( p j ) and q = ( q k ) be real sequences of nonnegative numbers with the property that P m = ∑ j = 0 m p j ≠ 0 and Q n = ∑ k = 0 n q k ≠ 0 for all m and n . Let ( P m ) and ( Q n ) be regulary varying positive indices. Assume that ( u m n ) is a double sequence of complex (real) numbers, which is ( N ¯ , p , q ; α , β ) summable with a finite limit, where ( α , β ) = ( 1,1 ) , ( 1,0 ) , or ( 0,1 ) . We present some conditions imposed on the weights under which ( u m n ) converges in Pringsheim's sense. These results generalize and extend the results obtained by authors in [Comput. Math. Appl., <strong>62</strong>, No. 6, 2609–2615 (2011)].