The Banach Algebra A* and Its Properties

Beurling's algebra A* = {f : ∑^∞_k=0sup_k≤|m| | f̂ (m)| < ∞ }is considered. A* arises quite naturally in problems of summability of the Fourier series at Lehesgue points, whereas Wiener's algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with [-π, π], and its dual space is indicated. Analogs of Herz's and Wiener-Ditkin's theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.