On the (dis)continuity of the Fourier transform of measures
Fourier transform of measures, translation bounded measures
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.
T. Spindeler, N. Strungaru. "On the (dis)continuity of the Fourier Transform of Measures", Journal of Mathematical Analysis and Applications 499(2),125062, 2021. DOI:10.1016/j.jmaa.2021.125062
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