Wiener Algebras and Trigonometric Series in a Coordinated Fashion

Let W(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 ∑k=-∞∞ckeikt is the Fourier series of an integrable function if and only if there exists a ϕ∈ W(R) such that ϕ(k) = c_k, k∈ Z. If f∈ W(R) , then the piecewise linear continuous function ℓ_f defined by ℓ_f(k) = f(k) , k∈ Z, belongs to W(R) as well. Moreover, ‖ℓf‖W0≤‖f‖W0. 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 are established.