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

R. M. Trigub

Izvestiya. Mathematics , Tome 17 (1981) no. 3, pp. 567-593.

Résumé

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.

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