Affiliation:
1. Sumy State University, Sumy, Ukraine
Abstract
New relations between the Banach algebras of absolutely convergent
Fourier integrals of complex-valued measures of Wiener and various issues of
trigonometric Fourier series (see classical monographs by A.~Zygmund [1]
and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new
properties of the Fourier series from the known properties of the
Wiener algebras, as well as new results to be obtained for those algebras
from the known properties of Fourier series. For example, criteria, i.e.
simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to
be a Fourier series, or the Fourier series of a function
of bounded variation, and so forth. Approximation properties of various
linear summability methods of Fourier series (comparison, approximation of
function classes and single functions) and summability almost everywhere
(often with the set indication) are considered.
The presented material was reported by the author on 12.02.2021 at the
Zoom-seminar on the theory of real variable functions at the Moscow State
University.
Publisher
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine
Reference49 articles.
1. Zygmund, A. (2003). Trigonometric Series. Cambridge University Press, Cambridge.
2. Bary, N.K. (1964). A Treatise on Trigonometric Series. Pergamon Press, New York.
3. Makarov, B.M., & Podkorytov, A.N. (2011). Lectures on Real Analysis. BHVPetersburg, St. Petersburg (in Russian).
4. Trigub, R., & Belinsky, E. (2004). Fourier Analysis and Approximation of Functions. Kluwer–Springer.
5. Liflyand, E., Samko, S., & Trigub, R. (2012). The Wiener algebra of absolutely convergent Fourier integrals: An overview. Anal. Math. Phys. 2 (1), 1–68. https://doi.org/10.1007/s13324-012-0025-6