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 Sibirsk. Mat. Zh., 2016, Volume 57, Number 1, Pages 186–198 (Mi smj2737)

On Wiener's Theorem for functions periodic at infinity

I. I. Strukova

Voronezh State University, Voronezh, Russia

Abstract: We consider the functions periodic at infinity with values in a complex Banach space. The notions are introduced of the canonical and generalized Fourier series of a function periodic at infinity. We prove an analog of Wiener's Theorem on absolutely convergent Fourier series for functions periodic at infinity whose Fourier series are summable with weight. The two criteria are given: for the function periodic at infinity to be the sum of a purely periodic function and a function vanishing at infinity and for a function to be periodic at infinity. The results of the article base on substantially use on spectral theory of isometric representations.

Keywords: Banach space, function slowly varying at infinity, function periodic at infinity, Fourier series, Wiener's Theorem.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00197 Russian Science Foundation 14-21-00066 The author was supported by the Russian Foundation for Basic Research (Grants 13-01-00378; 14-01-31196), the Russian Science Foundation (Grant 14-21-00066), and the Ministry of Science and Education in the framework of the State Tasks in Science to Institutions of Higher Education for 2014–2016 (Grant 1110).

DOI: https://doi.org/10.17377/smzh.2016.57.114

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English version:
Siberian Mathematical Journal, 2016, 57:1, 145–154

Bibliographic databases:

Document Type: Article
UDC: 517.9

Citation: I. I. Strukova, “On Wiener's Theorem for functions periodic at infinity”, Sibirsk. Mat. Zh., 57:1 (2016), 186–198; Siberian Math. J., 57:1 (2016), 145–154

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v prostranstvakh Stepanova”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:2 (2017), 172–182
2. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v odnorodnykh prostranstvakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 2(39), 29–38
3. A. G. Baskakov, I. I. Strukova, I. A. Trishina, “Solutions almost periodic at infinity to differential equations with unbounded operator coefficients”, Siberian Math. J., 59:2 (2018), 231–242
4. 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
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