On the (dis)continuity of the Fourier transform of measures

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:499

Article

Spindeler, Timo¹  Strungaru, Nicolae^2,3 

[1] Univ Alberta, Dept Math & Stat Sci, 632 CAB, Edmonton, AB T6G 2G1, Canada

[2] MacEwan Univ, Dept Math Sci, 10700-104 Ave, Edmonton, AB T5J 4S2, Canada

[3] Inst Math Simon Stoilow, Bucharest, Romania

关键词: Fourier transform of measures;    Translation bounded measures;    Positive definite measures;    Tempered distributions;    Vague topology for measures;   

DOI  :  10.1016/j.jmaa.2021.125062

来源: Elsevier

【 摘 要 】

In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set of measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free