Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus

In this paper the connections among three algebras are discussed: the algebra of Fourier transforms of finite Borel measures on $\mathbf R^m$, the algebra $A$ of absolutely convergent Fourier integrals, and the algebra of functions which generate a bounded multiplier sequence. Necessary and sufficient conditions for membership in $A$ are given, a Bernstein–Rogozinskii type of summation method for multiple Fourier series is investigated, and a comparison principle is formulated for various methods of summation of Fourier series according to their approximation properties. In addition, in connection with the well-known theorem of Jackson and its converse, various moduli of smoothness are introduced and studied.
Bibliography: 33 titles.