Conditions under which convergence of a sequence or its certain subsequences follows from deferred Ces\`{a}ro summability

Abstract

Let $(u_n: n= 1,2,...)$ be a sequence of real or complex numbers. We aim in this paper to determine necessary and/or sufficient conditions under which convergence of a sequence $(u_n)$ or its certain subsequences follows from summability by deferred Ces\`aro means. We also investigate the limiting behavior of deferred moving averages of $(u_n)$. The conditions in our theorems are one-sided if $(u_n)$ is a sequence of real numbers, and two-sided  if $(u_n)$ is a sequence of complex numbers. The theory developed in this paper should be useful for developing more interesting and useful results in connection with other sophisticated summability means as well as to extend to other spaces like ordered linear spaces.