Some Tauberian Theorems For Weighted Means of Double Integrals
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Tarih
2019
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Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Let p(x) and q(y) be nondecreasing continuous functions on [0; 1) such that p(0) = q(0) = 0 and p(x); q(y) ! 1 as x; y ! 1. For a locally integrable function f(x; y) on R2 + = [0; 1) [0; 1), we denote its double integral by F(x; y) = R x 0 R y 0 f(t; s)dtds and its weighted mean of type ( ; ) by t ;(x; y) = Z x 0 Z y 0 ? 1 p(t) p(x) ? ? 1 q(s) q(y) ?f(t; s)dtds where > 1 and > 1. We say that R 1 0 R 1 0 f(t; s)dtds is integrable to L by the weighted mean method of type ( ; ) determined by the functions p(x) and q(x) if limx;y!1 t ;(x; y) = L exists. We prove that if limx;y!1 t ;(x; y) = L exists and t ;(x; y) is bounded on R2 + for some > 1 and > 1, then limx;y!1 t +h;+k(x; y) = L exists for all h > 0 and k > 0. Finally, we prove that if R 1 0 R 1 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 0(x) Z y 0 f(x; s)ds = O(1) and q(y) q 0(y) Z x 0 f(t; y)dt = O(1) hold, then limx;y!1 F(x; y) = L exists.
Açıklama
Anahtar Kelimeler
0-Belirlenecek
Kaynak
Communications Series A1: Mathematics and Statistics
WoS Q Değeri
N/A
Scopus Q Değeri
Cilt
68
Sayı
2
