- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Vol: 68 Issue: 2
- Some Tauberian theorems for weighted means of double integrals
Some Tauberian theorems for weighted means of double integrals
Authors : Gökşen Findik, Ibrahim Çanak
Pages : 1452-1461
Doi:10.31801/cfsuasmas.539358
View : 6 | Download : 4
Publication Date : 2019-08-01
Article Type : Research
Abstract :Let p(x) and q(y) be nondecreasing continuous functions on [0,∞) such that p(0)=q(0)=0 and p(x),q(y)→∞ as x,y→∞. For a locally integrable function f(x,y) on R₊²=[0,∞)×[0,∞), we denote its double integral by F(x,y)=∫₀^{x}∫₀^{y}f(t,s)dtds and its weighted mean of type (α,β) by t_{α,β}(x,y)=∫₀^{x}∫₀^{y}(1-((p(t))/(p(x))))^{α}(1-((q(s))/(q(y))))^{β}f(t,s)dtds where α>-1 and β>-1. We say that ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (α,β) determined by the functions p(x) and q(x) if lim_{x,y→∞}t_{α,β}(x,y)=L exists. We prove that if lim_{x,y→∞}t_{α,β}(x,y)=L exists and t_{α,β}(x,y) is bounded on R₊² for some α>-1 and β>-1, then lim_{x,y→∞}t_{α+h,β+k}(x,y)=L exists for all h>0 and k>0. Finally, we prove that if ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (1,1) determined by the functions p(x) and q(x) and conditions [displaystyle]Keywords : Divergent integrals, weighted means of double integrals, Tauberian theorems and conditions