Some Tauberian conditions under which convergence follows from (C, 1, 1, 1) summability
Articles > Article · 2019
Authors: ÜMİT TOTUR, İbrahim ÇANAK
The Journal of Analysis, Refereed Journals of Other Institutions, International, Scopus, 2019, vol 28, pp 683-694
DOI: 10.1007/s41478-019-00179-z
Summary
Given a real-valued integrable function f(x, y, z) which is integrable over [0,infinity )x[0,infinity )x[0,infinity), let s(x, y, z) denote its integral over [0,x]x[0,y]x[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]x[0,y]x[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 Moricz (Stud Math 138(1):41-52, 2000) for improper double integrals to improper triple integrals.