General Tauberian theorems for the Cesàro integrability of functions
Articles > Article · 2019
Authors: ÜMİT TOTUR, İbrahim ÇANAK
Georgian Mathematical Journal, Journal Indexed in SCI Expanded, International, WoS, 2019, vol 27, pp 517-527
Summary
Abstract For a locally integrable function f on [ 0 , ∞ ) {[0,\infty)} , we define F ( t ) = ∫ 0 t f ( u ) d u and σ α ( t ) = ∫ 0 t ( 1 - u t ) α f ( u ) d u F(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!du for t > 0 {t>0} . The improper integral ∫ 0 ∞ f ( u ) d u {\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be (