Necessary and sufficient Tauberian conditions under which convergence of double sequences follows from Ar,δ summability

Abstract

Let x = (xmn) be a double sequence of real or complex numbers. The Ar,δ-transform of a sequence (xmn) is defined by (Ar,δx)mn=σmnr,δ(x)=1(m+1)(n+1)∑j=0m∑k=0n(1+rj)(1+δk)xjk, 0<r,δ<1We say that (xmn) a sequence is (Ar,δ,1,1) summable to l if (σmnr,δ(x)) has a finite limit l. It is known that if limm,n→∞xmn=l and (xmn) is bounded, then the limit limm,n→∞σmnr,δ(x)=l exists. But the inverse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for (Ar,δ,1,1) summability method under which the inverse of this implication holds. Following Tauberian theorems for (Ar,δ, 1, 1) summability method, we also define Ar and Aδ transformations of double sequences and obtain Tauberian theorems for the (Ar,δ, 1, 0) and (Ar,δ, 0, 1) summabillity methods.