SOME TAUBERIAN THEOREMS FOR WEIGHTED MEANS OF DOUBLE INTEGRALS
Makaleler > Tam Makale · 2019
Yazarlar: İbrahim ÇANAK, Ibrahim Canak
COMMUNICATIONS FACULTY OF SCIENCES UNIVERSITY OF ANKARA-SERIES A1 MATHEMATICS AND STATISTICS, SCI Expanded İndekslerine Giren Dergi, Uluslararası, Wos, 2019, cilt 68, ss 1452-1461
DOI: 10.31801/cfsuasmas.539358
Özet
Let p(x) and q(y) be nondecreasing continuous functions on [0, infinity) such that p(0) = q(0) = 0 and p(x), q(y) -> infinity as x, y -> infinity. For a locally integrable function f(x,y) on R-+(2) = [0, infinity) x [0, infinity), we denote its double integral by F(x, y) = integral(x)(0) integral(y)(0) f (t, s)dtds and its weighted mean of type (alpha, beta) byt(alpha,beta) (x,y) = integral(x)(0) integral(y)(0) (1-p(t)/p(x))(alpha) (1-q(s)/q(y))(beta) f(t,s)dtdswhere alpha > -1 and beta > -1. We say that integral(infinity)(0) integral(infinity)(0) f (t, s) dtds is integrable to L by the weighted mean method of type (alpha, beta) determined by the functions p(x) and q(x) if lim(x,y) (-> infinity )t(alpha,beta) (x,y) = L exists. We prove that if lim(x,y) (-> infinity )t(alpha,beta) (x,y) = L exists and t(alpha,beta) (x,y) is bounded on R-+(2) for some alpha > -1 and beta > - 1, then lim(x,y) (-> infinity )t(alpha+h, beta+k) (x,y) = L exists for all h > 0 and k > 0. Finally, we prove that if integral(infinity)(0) integral(infinity )(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 p(x)/p'(x) integral(y)(0) f (x, s)ds = O(1) and q(y)/q'(y) integral(x)(0) f(t,y)dt = O(1) hold, then lim(x,y) (-> infinity) F(x, y) = L exists.