Abstract
We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.
Highlights
Convergence of the Fourier series and integral of integrable functions of one variable at certain point depends only from the values of the function in the small neighbourhood of this point
In [2] it is given a review of recent results on equiconvergence of expansions in multiple trigonometric Fourier series and integral in the case of summation over rectangles
In [7] a comparison theorem on equiconvergence of the Fourier Jacobi series with certain trigonometric Fourier series is proved
Summary
Convergence of the Fourier series and integral of integrable functions of one variable at certain point depends only from the values of the function in the small neighbourhood of this point (localizations principles). The difference of the partial sums of the Fourier series and integral of a function uniformly converge to zero, which means both expansions converge or diverge at the same time (equiconvergence). In [3] the problem of equiconvergence for expansions in a triple trigonometric Fourier series and a Fourier integral of continuous functions with a certain modulus of continuity in the case of a lacunary sequence of partial sums is studied. In this paper we study equiconvergence of the Fourier series and integral of the linear continuous functionals (distributions) in the case of spherical summation. We will prove uniform equiconvergence of the Riesz means of the Fourier series and the Fourier integral expansion.
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