Some Tauberian conditions under which convergence follows from (C, 1, 1, 1) summability

Abstract

Given a real-valued integrable function f(x, y, z) which is integrable over $[0,\infty )\times [0,\infty )\times [0,\infty )$, let s(x, y, z) denote its integral over $[0,x]\times [0,y]\times [0,z]$ and let $\sigma (x,y,z)$ denote its (C, 1, 1, 1) mean, the average of s(x, y, z) over $[0,x]\times [0,y]\times [0,z]$, where $x,y,z >0.$ We give one-sided Tauberian conditions of Landau and Hardy type under which convergence of s(x, y, z) follows from (C, 1, 1, 1) summability of s(x, y, z). We obtain convergence of s(x, y, z) from its (C, 1, 1, 1) summability provided that s(x, y, z) is slowly oscillating in certain senses. Furthermore, we extend a Tauberian theorem given by Móricz (Stud Math 138(1):41–52, 2000) for improper double integrals to improper triple integrals.