# On Fourier Series on the Torus and Fourier Transforms

Authors
• 1 Donetsk National University, Donetsk, 83114, Ukraine , Donetsk (Ukraine)
Type
Published Article
Journal
Mathematical Notes
Publisher
Publication Date
Nov 01, 2021
Volume
110
Issue
5-6
Pages
767–772
Identifiers
DOI: 10.1134/S0001434621110134
Source
Springer Nature
Keywords
Disciplines
• article
Abstract The question of the representability of a continuous function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^d$$\end{document} in the form of the Fourier integral of a finite Borel complex-valued measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^d$$\end{document} is reduced in this article to the same question for a simple function. This simple function is determined by the values of the given function on the integer lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^d$$\end{document}. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document}, this result is already known: it is an inscribed polygonal line. The article also describes applications of the obtained theorems to multiple trigonometric Fourier series.