Tauberian theorems for the Cesàro summability method of regularly generated double integrals

Abstract

For a continuous function 1 over R2+ := [1,?) ? [1,?), we denote its integral over [1, x] ? [1, y] by h(x, y) = R x 1 R y 1 1(u, v)dudv and its (C, 1, 1) mean, the average of h(x, y) over [1, x] ? [1, y], by t(h(x, y)) = (xy)?1 R x 1 R y 1 h(u, v)dudv. Analogously, the other means (C, 1, 0) and (C, 0, 1) can be defined. In this paper, we introduce the concept of regularly generated double integrals in senses (1, 1), (1, 0) and (0, 1) and obtain Tauberian conditions in terms of the regularly generated double integrals in senses (1,1), (1,0) and (0,1) under which convergence of h(x, y) follows from that of t(h(x,y)).