Wiener algebras and trigonometric series in a coordinated fashion

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 $\sum _{k=-\infty }^{\infty }{c}_k{e}^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $\varphi \in W_0(\mathbb{R})$ such that $\varphi (k)=c_k$, $k\in \mathbb{Z}$. 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, $\Vert {\ell }_f{\Vert }_{W_0}\le \Vert f{\Vert }_{W_0}$. 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.

20 pages