Tauberian theorems for the Ces\`aro summability method of regularly generated double integrals

Abstract

For a locally integrable function $g$ over $\mathbb{R}_{+}^{2}:=
[1,\infty)\times[1,\infty)$, we denote its integral over $[1,x]\times[1,y]$ by $h(x,y)=\int_{1}^{x}\int_{1}^{y}g(u,v)dudv$ and its $(C,1,1)$ mean, the average of $h(x,y)$ over $[1,x]\times[1,y]$, by $t(h(x,y))=(xy)^{-1}\int_{1}^{x}\int_{1}^{y}h(u,v)dudv$. 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 $t(h(x,y))$.