  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.
