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 Probl. Anal. Issues Anal., 2018, Volume 7(25), Issue 2, Pages 3–19 (Mi pa244)

On the almost periodic at infinity functions from homogeneous spaces

A. G. Baskakov, V. E. Strukov, I. I. Strukova

Voronezh State University, 1 Universitetskaya sq., Voronezh 394006, Russia

Abstract: We consider homogeneous spaces of functions defined on the real axis (or semi-axis) with values in a complex Banach space. We study the new class of almost periodic at infinity functions from homogeneous spaces. The main results of the article are connected to harmonic analysis of those functions. We give four definitions of an almost periodic at infinity function from a homogeneous space and prove them to be equivalent. We also introduce the concept of a Fourier series with slowly varying at infinity coefficients (neither necessarily constant nor necessarily having a limit at infinity). It is proved that the Fourier coefficients of almost periodic at infinity function from a homogeneous space (not necessarily continuous) can be chosen continuous. Moreover, they can be extended on $\mathbb{C}$ to bounded entire functions of exponential type. Besides, we prove the summability of Fourier series by the method of Bochner–Fejer. The results were received with essential use of isometric representations and Banach modules theory.

Keywords: almost periodic at infinity function, homogeneous space, Banach module, almost periodic vector, Fourier series.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00197_à18-31-00097 The first author of this work was supported by RFBR according to the research project 16-01-00197, the second author was supported by RFBR according to the research project 18-31-00097 and the third author was supported by RFBR according to the research project 18-31-00097.

DOI: https://doi.org/10.15393/j3.art.2018.4990

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Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: 33E30, 43A60, 46B04
Revised: 03.12.2018
Language: English

Citation: A. G. Baskakov, V. E. Strukov, I. I. Strukova, “On the almost periodic at infinity functions from homogeneous spaces”, Probl. Anal. Issues Anal., 7(25):2 (2018), 3–19

Citation in format AMSBIB
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