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 Rm, 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-Rogosinski 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.