Wiener algebras and trigonometric series in a coordinated fashion

TL;DR

This paper characterizes when trigonometric series are Fourier series of integrable functions using Wiener algebras, establishing new conditions and properties with various applications.

Contribution

It introduces novel criteria linking Wiener algebras to Fourier series, expanding understanding of their structure and applications.

Findings

Characterization of Fourier series via Wiener algebra functions

Piecewise linear functions belong to Wiener algebra with controlled norm

New necessary and sufficient conditions for trigonometric series to be Fourier series

Abstract

Let $W_0(\mathbb R)$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $\sum\limits_{k=-\infty}^\infty c_k e^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $\phi\in W_0(\mathbb R)$ such that $\phi(k)=c_k$, $k\in\mathbb Z$. If $f\in W_0(\mathbb R)$, then the piecewise linear continuous function $\ell_f$ defined by $\ell_f(k)=f(k)$, $k\in\mathbb Z$, belongs to $W_0(\mathbb R)$ as well. Moreover, $\|\ell_f\|_{W_0}\le \|f\|_{W_0}$. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of $W_0$ are established.