On Fourier Series on the Torus and Fourier Transforms

Abstract

The question of the representability of a continuous function on $\mathbb R^d$ in the form of the Fourier integral of a finite Borel complex-valued measure on $\mathbb R^d$ 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 $\mathbb R^d$. For $d=1$, 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.