Strong Cesàro summability and statistical limit of fourier integrals

Abstract

Abstract In our previous paper [2], we introduced the concept of statistical limit of a measurable function and that of strong Cesàro summability of a locally integrable function. As an application, we proved there that the Fourier integral of a function ƒ ∈ L1(ℝ)∩C(ℝ) is strongly Cesàro summable and has statistical limit at every point of ℝ. The purpose of this paper is twofold: (i) We prove that if ƒ ∈ L1(ℝ) is locally bounded on ℝ and continuous on an open interval J , then the strong Cesàro summability and the existence of the statistical limit of the Fourier integral of ƒ is uniform on J. (ii) We complete and simplify the proof of [2, Statement (γ) of Theorem 3], which says that if ƒ ∈ L1(ℝ)∩C0(ℝ), then the conclusion in (i) is uniform on the whole ℝ.