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.