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