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.
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