Some Tauberian theorems for weighted means of double integrals

Authors : Gökşen FINDIK, İbrahim ÇANAK

Pages : 1452-1461

Doi:10.31801/cfsuasmas.539358

View : 9 | Download : 5

Publication Date : 2019-08-01

Article Type : Research Paper

Abstract :Let pinsert ignore into journalissuearticles values(x); and qinsert ignore into journalissuearticles values(y); be nondecreasing continuous functions on [0,∞); such that pinsert ignore into journalissuearticles values(0);=qinsert ignore into journalissuearticles values(0);=0 and pinsert ignore into journalissuearticles values(x);,qinsert ignore into journalissuearticles values(y);→∞ as x,y→∞. For a locally integrable function finsert ignore into journalissuearticles values(x,y); on R₊²=[0,∞);×[0,∞);, we denote its double integral by Finsert ignore into journalissuearticles values(x,y);=∫₀^{x}∫₀^{y}finsert ignore into journalissuearticles values(t,s);dtds and its weighted mean of type insert ignore into journalissuearticles values(α,β); by t_{α,β}insert ignore into journalissuearticles values(x,y);=∫₀^{x}∫₀^{y}insert ignore into journalissuearticles values(1-insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(pinsert ignore into journalissuearticles values(t););/insert ignore into journalissuearticles values(pinsert ignore into journalissuearticles values(x););););^{α}insert ignore into journalissuearticles values(1-insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(qinsert ignore into journalissuearticles values(s););/insert ignore into journalissuearticles values(qinsert ignore into journalissuearticles values(y););););^{β}finsert ignore into journalissuearticles values(t,s);dtds where α>-1 and β>-1. We say that ∫₀^{∞}∫₀^{∞}finsert ignore into journalissuearticles values(t,s);dtds is integrable to L by the weighted mean method of type insert ignore into journalissuearticles values(α,β); determined by the functions pinsert ignore into journalissuearticles values(x); and qinsert ignore into journalissuearticles values(x); if lim_{x,y→∞}t_{α,β}insert ignore into journalissuearticles values(x,y);=L exists. We prove that if lim_{x,y→∞}t_{α,β}insert ignore into journalissuearticles values(x,y);=L exists and t_{α,β}insert ignore into journalissuearticles values(x,y); is bounded on R₊² for some α>-1 and β>-1, then lim_{x,y→∞}t_{α+h,β+k}insert ignore into journalissuearticles values(x,y);=L exists for all h>0 and k>0. Finally, we prove that if ∫₀^{∞}∫₀^{∞}finsert ignore into journalissuearticles values(t,s);dtds is integrable to L by the weighted mean method of type insert ignore into journalissuearticles values(1,1); determined by the functions pinsert ignore into journalissuearticles values(x); and qinsert ignore into journalissuearticles values(x); and conditions [displaystyle]\displaystyleinsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(pinsert ignore into journalissuearticles values(x););/insert ignore into journalissuearticles values(p′insert ignore into journalissuearticles values(x);););∫₀^{y}finsert ignore into journalissuearticles values(x,s);ds=Oinsert ignore into journalissuearticles values(1); and [displaystyle]\displaystyleinsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(qinsert ignore into journalissuearticles values(y););/insert ignore into journalissuearticles values(q′insert ignore into journalissuearticles values(y);););∫₀^{x}finsert ignore into journalissuearticles values(t,y);dt=Oinsert ignore into journalissuearticles values(1); hold, then lim_{x,y→∞}Finsert ignore into journalissuearticles values(x,y);=L exists.
Keywords : Divergent integrals, weighted means of double integrals, Tauberian theorems and conditions