Abstract
Beurling’s algebra\(A^* = \{ f:\sum\nolimits_{k = 0}^\infty {\sup _{k \leqslant |m|} |\hat f(m)|< \infty } \} \) 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belinskii, E.S. (1975). Summability of multiple Fourier series in Lebesgue points.Theory of Functions, Functional Analysis and Their Applications 23, 3–12. (Russian)
Trigub, R. M. (1983). Absolute convergence of Fourier integrals and approximation of functions by the linear means of their Fourier series.Constructive Function Theory’81, Sofia. 178–180. (Russian)
—————. (1989). Multipliers of Fourier series and approximation of functions by polynomials in spacesC andL.Dokl. Akad. Nauk SSSR 306, 292–296; English transl.,Soviet Math. Dokl. 39, 494–498.
Telyakovskii, S. A. (1973). On a sufficient condition of Sidon for the integrability of trigonometric series.Mat. Zametki 14, 317–328; English transl.,Math. Notes Acad. Sci. USSR 14, 742–748.
Wiener, N. (1933).The Fourier Integral and Certain of Its Applications. Cambridge University Press, Cambridge.
Zygmund, A. (1959).Trigonometric Series, Vol. I, II. Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Belinskii, E.S., Liflyand, E.R. & Trigub, R.M. The banach algebra A and its properties. The Journal of Fourier Analysis and Applications 3, 103–129 (1997). https://doi.org/10.1007/BF02649131
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02649131