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 Eurasian Math. J., 2016, Volume 7, Number 4, Pages 9–29 (Mi emj238)

Harmonic analysis of functions periodic at infinity

Faculty of Applied Mathematics, Mechanics and Informatics, Voronezh State University, 1 Universitetskaya Sq, 394036 Voronezh, Russia

Abstract: In this paper we introduce the notion of vector-valued functions periodic at infinity. We characterize the sums of the usual periodic functions and functions vanishing at infinity as a subclass of these functions. Our main focus is the development of the basic harmonic analysis for functions periodic at infinity and an analogue of the celebrated Wiener’s Lemma that deals with absolutely convergent Fourier series. We also derive criteria of periodicity at infinity for solutions of difference and differential equations. Some of the results are derived by means of the spectral theory of isometric group representations.

Keywords and phrases: Banach space, functions slowly varying at infinity, functions periodic at infinity, Wiener's theorem, absolutely convergent Fourier series, invertibility, difference equations.

 Funding Agency Grant Number Russian Science Foundation 14-21-00066 Russian Foundation for Basic Research 16-01-00197_a The results of Section 5 were obtained with support of the Russian Science Foundation, project no. 14-21-00066 in the Voronezh State University. The other results were obtained with support of the Russian Foundation for Basic Research, project no. 16-01-00197 in the Voronezh State University.

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Document Type: Article
MSC: 34A55, 34B05, 58C40
Language: English

Citation: A. Baskakov, I. Strukova, “Harmonic analysis of functions periodic at infinity”, Eurasian Math. J., 7:4 (2016), 9–29

Citation in format AMSBIB
\Bibitem{BasStr16} \by A.~Baskakov, I.~Strukova \paper Harmonic analysis of functions periodic at infinity \jour Eurasian Math. J. \yr 2016 \vol 7 \issue 4 \pages 9--29 \mathnet{http://mi.mathnet.ru/emj238} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000398292700001} 

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This publication is cited in the following articles:
1. 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
2. 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
3. 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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