Conditions for the pringsheim convergence of double sequences that are deferred cesàro summable

Abstract

For a given real or complex valued double sequence (umn), its deferred Cesàro means are defined by Dmn(11)(u)=1(βm−αm)(qn−pn)∑j=αm+1βm∑k=pn+1qnujk(1) where (pn), (qn), (αm) and (βm) are the sequences of non-negative integers satisfying pn < qn, αm < βm and limn qn = ∞, limm βm = ∞. We say that (umn) is deferred Cesàro summable (briefly (DC, 1, 1) summable) to l if (1) tends to l as m, n → ∞. Note that, if Pn = 0, qn = n and αm = 0, βm = m, then corresponding (DC, 1, 1) method is the well known Cesàro summability (C, 1, 1).In this extended abstract we give inverse conditions to obtain Pringsheim convergence of deferred Cesàro summable double sequences. We also give an inclusion relation with example.