Some properties of Fourier integrals

Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely convergent Fourier series. New properties of functions from these algebras are obtained. Some conditions which determine membership of f in F(R) are given. For many elementary functions f the problem of belonging f to F(R) can be resolved easily using these conditions. We prove that the Hilbert operator is a bijective isometric operator in the Banach spaces W_0, F(R), K(R)-A_1 (A_1 is the one-dimension space of constant functions). We also consider the classes M_k, which are similar to the Bochner classes F_k, and obtain integral representation of the Carleman transform of measures of M_k by integrals of some specific form.