On Fourier Series on the Torus and Fourier Transforms

R. M. Trigub

Matematičeskie zametki, Tome 110 (2021) no. 5, pp. 766-772

Résumé

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.

Détail

Keywords: Fourier series of a measure on the torus $\mathbb T^d$ and functions from $L_1(\mathbb T^d)$, variation of a measure, Wiener Banach algebras, positive definite functions, exponential entire functions, $(C,1)$-means of Fourier series, Banach–Alaoglu theorem.
Mots-clés : Vitali variation