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 proved in the paper that, in particular, a trigonometric series $\sum \nolimits _{k=-\infty }^\infty c_k e^{ikt}$ 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, $\Vert \ell _f\Vert _{W_0}\le \Vert f\Vert _{W_0}$. 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.