Harmonic analysis of periodic at infinity functions in homogeneous spaces

I. I. Strukova

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2017), pp. 29-38.

Résumé

This article is devoted to homogeneous spaces $\mathrm{F}(\mathrm{R},X)$ of functions defined on $\mathrm{R}$ with their values in a complex Banach space X. We introduce a notion of slowly varying at infinity function from $\mathrm{F}(\mathrm{R},X)$. We also consider some criteria for a function to be slowly varying at infinity. Then it is stated that for each slowly varying at infinity function from any homogeneous space (not necessary continuous, for instance, a function from Stepanov space $S^p(R, X)$ or $L^p(R, X)$, ) there exists a uniformly continuous slowly varying at infinity function that differs from the first one by a function decreasing at infinity. In other words, a function from the corresponding subspace $F_0(\mathrm{R}, X)$ . In the second part of the article we introduce a notion of periodic at infinity function from homogeneous space. Our main results are connected with harmonic analysis of periodic at infinity functions from $F(\mathrm{R}, X)$. Periodic at infinity functions appear naturally as bounded solutions of certain classes of differential and difference equations. In this paper we develop basic harmonic analysis for such functions. We introduce the notion of a generalized Fourier series of a periodic at infinity function from homogeneous space. The Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity. Moreover, it is stated that generalized Fourier coefficients of a function that may not be continuous can be chosen continuous. We use methods of the spectral theory of locally compact Abelian group isometric representations (Banach modules over group algebras).

Détail

Keywords: Banach space, homogeneous space, slowly varying at infinity function, periodic at infinity function, Fourier series.
Mots-clés : $L^1(\mathrm{R})$-module

Bibliographie

[1] A.\;G. Baskakov, I.\;A. Krishtal, “Harmonic Analysis of Causal Operators and Their Spectral Properties”, Izv. RAN. Ser. matematicheskaya, 69:3 (2005), 3–54 | DOI | MR | Zbl

[2] A.\;G. Baskakov, “Harmonic and Spectral Analysis of Power Bounded Operators and Bounded Semigroups of Operators on Banach Spaces”, Mat. zametki, 97:2 (2015), 174–190 | DOI | Zbl

[3] A.\;G. Baskakov, “Analysis of Linear Differential Equations by Methods of the Spectral Theory of Difference Operators and Linear Relations”, UMN, 68:1 (2013), 77–128 | DOI | MR | Zbl

[4] A.\;G. Baskakov, “Bernšte n-Type Inequalities in Abstract Harmonic Analysis”, Sib. mat. zhurn., 20:5 (1979), 942–952 | MR | Zbl

[5] A.\;G. Baskakov, “On Spectral Synthesis in Banach Modules over Commutative Banach Algebras”, Mat. zametki, 34:4 (1983), 573–585 | MR | Zbl

[6] A.\;G. Baskakov, “Spectral Analysis of Differential Operators with Unbounded Operator-Valued Coefficients, Difference Relations and Semigroups of Difference Relations”, Izv. RAN. Ser. matematicheskaya, 73:2 (2009), 3–68 | DOI | MR | Zbl

[7] A.\;G. Baskakov, “Spectral Tests for the Almost Periodicity of the Solutions of Functional Equations”, Mat. zametki, 24:2 (1978), 195–206

[8] A.\;G. Baskakov, “Beurling’s Theorem for Functions with Essential Spectrum from Homogeneous Spaces and Stabilization of Solutions of Parabolic Equations”, Mat. zametki, 92:5 (2012), 643–661 | DOI | MR | Zbl

[9] A.\;G. Baskakov, “Theory of Representations of Banach Algebras, and Abelian Groups and Semigroups in the Spectral Analysis of Linear Operators”, SMFN, 9 (2004), 3–151 | Zbl

[10] N. Wiener, Fourier Integral and Some of Its Applications, Nauka Publ., Moscow, 1963, 256 pp.

[11] Yu.\;L. Daletskiy, M.\;G. Kreyn, Stability of Solutions of Differential Equations in Banach Space, Nauka Publ., Moscow, 1970, 535 pp. | MR

[12] B.\;M. Levitan, Zhikov\;V. V., Almost Periodic Functions and Differential Equations, Izd-vo MGU, Moscow, 1978, 205 pp. | MR

[13] I.\;I. Strukova, “Harmonic Analysis of Periodic Vectors and Functions Periodic at Infinity”, Vestn. NGU. Ser.: mat., mekh., inform., 14:1 (2014), 98–111 | Zbl

[14] I.\;I. Strukova, “On the Harmonic Analysis of Periodic at Infinity Functions”, Izv. Sarat. univ. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:1 (2014), 28–38 | Zbl

[15] I.\;I. Strukova, “On the Wiener Theorem for Periodic at Infinity Functions”, Sib. mat. zhurn., 57:1 (2016), 186–198 | MR | Zbl

[16] I.\;I. Strukova, “Spectra of Algebras of Slowly Varying and Periodic at Infinity Functions and Banach Limits”, Vestn. VGU. Seriya: Fizika. Matematika, 2015, no. 3, 161–165 | Zbl

[17] I.\;I. Strukova, “Wiener’s Theorem for Periodic at Infinity Functions”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 12:4 (2012), 34–41 | Zbl

[18] I.\;I. Strukova, “Wiener’s Theorem for Periodic at Infinity Functions with Summable Weighted Fourier Series”, Ufim. mat. zhurn., 5:3 (2013), 144–152 | MR

[19] E. Hille, R.\;S. Phillips, Functional Analysis and Semi-Groups, IL Publ., Moscow, 1962, 829 pp. | MR

[20] A. Baskakov, I. Strukova, “Harmonic analysis of functions periodic at infinity”, Eurasian Math. J., 7:4 (2016), 7–26 | MR

[21] K.\;-J. Engel, R. Nagel, A short course on operator semigroups, Universitext, Springer, New York, 2006, 247 pp. | MR | Zbl