The banach algebra A and its properties |
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Authors: | E S Belinskii E R Liflyand R M Trigub |
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Institution: | (1) Department of Mathematics, University of Zimbabwe, Mount Pleasant, P.O. Box MP 167, Harare, Zimbabwe;(2) Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel;(3) Department of Mathematics, Donetsk State University, ul. Universitetskaya 24, 340055 Donetsk, Ukraine |
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Abstract: | Beurling’s algebra
is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue 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. |
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Keywords: | Primary 42A28 42A24 42A16 |
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