Convergence follows from Ces\`{a}ro summability in the case of slowly decreasing or slowly oscillating double sequences in certain senses

Abstract

Let $(u_{\mu\nu})$ be a double sequence of real or complex numbers which is $(C, 1, 1)$ summable to a finite limit. We obtain some Tauberian conditions of slow decreasing or oscillating types in terms of the generator sequences in certain senses under which $P$-convergence of a double sequence $(u_{\mu\nu})$ follows from its $(C,1,1)$ summability. We give Tauberian theorems in which Tauberian conditions are of Hardy and Landau types as special cases of our results. We present some Tauberian conditions in terms of the de la Vall\'{e}e Poussin means of double sequences under which $P$-convergence of a double sequence $(u_{\mu\nu})$ follows from its $(C,1,1)$ summability. Moreover, we give analogous results for $(C,1,0)$ and $(C,0,1)$ summability methods.