Tauberian conditions under which convergence follows from Ces\`{a}ro summability of double integrals over $\mathbb{R}_{+}^{2}$

Abstract

For a real- or complex-valued valued continuous function $f$ over $\mathbb{R}_{+}^{2}:=[0,\infty)\times[0,\infty)$, we denote its integral over $[0,u]\times[0,v]$ by $s(u,v)$ and its $(C,1,1)$ mean, the average of $s(u,v)$ over $[0,u]\times[0,v]$, by $\sigma(u,v)$. The other means $(C,1,0)$ and $(C,0,1)$ are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over $\mathbb{R}_{+}^{2}$. We give one-sided and two-sided Tauberian conditions based on the difference between double integral of $s(u,v)$ and its means in different senses for Ces\`{a}ro summability methods of double integrals over $[0,u]\times[0,v]$ under which convergence of $s(u,v)$ follows from integrability of $s(u,v)$ in different senses.