Conditions fo the Pringsheim Convergence of Double Sequences That Are Deferred Cesaro Summable

Abstract

For a given real or complex valued double sequence (u(mn)), its deferred Cesaro means are defined by D-mn((11)) (u) = 1/(beta(m) - alpha(m))(q(n) -p(n)) Sigma(beta m)(j=alpha m+1) Sigma(qn)(k=Pn+1) u(jk) (1) where (p(n)), (q(n)), (alpha(m)) and (beta(m)) are the sequences of non-negative integers satisfying p(n) < q(n), alpha(m) < beta(m) and lim(n) q(n) = infinity, lim(m) beta(m) = infinity. We say that (u(mn)) is deferred Ces`aro summable (briefly ( DC, 1, 1) summable) to l if (1) tends to l as m, n -> infinity. Note that, if p(n) = 0, q(n) = n and alpha(m) = 0, beta(m) = m, then corresponding (DC, 1, 1) method is the well known Ces`aro summability (C, 1, 1). In this extended abstract we give inverse conditions to obtain Pringsheim convergence of deferred Ces`aro summable double sequences. We also give an inclusion relation with example.