Local versions of the Wiener–Lévy theorem

  • S. Yu. Favorov Karazin National University of Kharkiv
Keywords: Wiener–Lévy theorem; Fourier transform; absolute convergent Diriclet series; pure point measure; real-analytic function.

Abstract

Let $h$ be a real-analytic function on the neighborhood of some compact set $K$ on the plane, and let $f(y)$ be the Fourier--Stieltjes transform of a complex measure of a finite total variation without singular components on the Euclidean space. Then there exists another measure of a finite total variation with the Fourier--Stieltjes transform $g(y)$ such that $g(y)=h(f(y))$ whenever the value $f(y)$ belongs to $K$.

Author Biography

S. Yu. Favorov, Karazin National University of Kharkiv

Kharkiv National University, Svobody sq, 4, faculty of mechanics and mathematics

References

N.I. Akhiezer, Theory of Approximation, F. Ungar Pub., 1956.

S.Yu. Favorov, Large Fourier quasicryals and Wiener’s theorem, Journal of Fourier Analysis and Applications, 25 (2019), No2, 377–392.

S.Yu. Favorov, Local Wiener’s theorem and Coherent sets of frequencies, Analysis Math., 46, (2020), No4, 737–746.

S.Yu. Favorov, Temperate distributions with locally finite support and spectrum on Euclidean spaces, arXiv:2106.07073, to appear in: Israel Journal of Mathematics.

M. Baake, R. Moody, Directions in mathematical quasicrystals, eds. CRM Monograph series, 2000, V.13, AMS, Providence RI, 379 p.

J.C. Lagarias, Geometric models for quasicrystals I. Delone set of finite type, Discr. and Comp. Geometry, 21 (1999) 161–191.

Y. Meyer Guinand’s measure are almost periodic distributions, Bulletin of the Hellenic Mathematical Society, 61, (2017) 11–20.

H. Reiter, J.D. Stegeman, Classical harmonic analysis and locally compact groups, Oxford University Press, Oxford, 2000.

W. Rudin, Fourier analysis on groups. interscience publications, a division of John Wiley and Sons, New York, 1962.

W. Rudin, Functional analysis, McGraw-Hill Book Company, New York, 1973.

A. Zygmund, Trigonometric series, Cambridge Univesity Press, Cambridge, 2002.

Published
2022-03-31
How to Cite
Favorov, S. Y. (2022). Local versions of the Wiener–Lévy theorem. Matematychni Studii, 57(1), 45-52. https://doi.org/10.30970/ms.57.1.45-52
Section
Articles