Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Strukova I. I. Harmonic Analysis of Periodic at Infinity Functions from Stepanov Spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 2, pp. 172-182. DOI: 10.18500/1816-9791-2017-17-2-172-182, EDN: ZEVXAL

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
22.05.2017
Full text:
(downloads: 184)
Language: 
Russian
Heading: 
UDC: 
517.9
EDN: 
ZEVXAL

Harmonic Analysis of Periodic at Infinity Functions from Stepanov Spaces

Autors: 
Strukova Irina Igorevna, Voronezh State University
Abstract: 

We consider Stepanov spaces of functions defined on R with their values in a complex Banach space. We introduce the notions of slowly varying and periodic at infinity functions from Stepanov space. The main results of the article are concerned with harmonic analysis of periodic at infinity functions from Stepanov space. For this class of functions we introduce the notion of a generalized Fourier series; the Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity. We prove analogs of the classical results on Cesaro summability. Basic results are derived with the use of isometric representations theory.

References: 
  1. Levitan B. M., Zhikov V. V. Almost periodic functions and differential equations. Cambridge, Cambridge Univ. Press, 1983. 224 p. (Russ. ed. : Levitan B. M., Zhikov V. V. Pochti-periodicheskye funkcii i differencialnye uravneniya. Moscow, Moscow. Univ. Press, 1978. 206 p.)
  2. Baskakov A. G. Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations. Russian Math. Surveys, 2013, vol. 68, no. 1, pp. 69–116. DOI: https://doi.org/10.4213/rm9505.
  3. Baskakov A. G. Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations. Izv. Math. 2009. vol. 73, no. 2, pp. 215 278. DOI: https://doi.org/10.4213/im2643.
  4. Baskakov A. G. Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators. J. Math. Sci., 2006, vol. 137, iss. 4, pp. 4885–5036. DOI: https://doi.org/10.1007/s10958-006-0286-4.
  5. Baskakov A. G. Harmonic analysis of linear operators. Voronezh, Voronezh Univ. Press, 1987. 165 p. (in Russian).
  6. Baskakov A. G., Krishtal I. A. Harmonic analysis of causal operators and their spectral properties. Izv. Math., 2005, vol. 69, no. 3, pp. 439–486. DOI: https://doi.org/10.4213/im639.
  7. Baskakov A. G. Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces. Math. Notes, 2015, vol. 97, no. 2, pp. 164–178. DOI: https://doi.org/10.4213/mzm10285.
  8. Baskakov A. G. Spectral tests for the almost periodicity of the solutions of functional equations. Math. Notes, 1978, vol. 24, no. 1–2, pp. 606–612.
  9. Baskakov A. G. Bernˇ ste˘ ın-type inequalities in abstract harmonic analysis. Siberian Math. J., 1979, vol. 20, no. 5, pp. 665–672.
  10. Hille E., Phillips R. Functional analysis and semi-groups. Amer. Math. Soc. Colloquim Publ., vol. 31. R. I., Amer. Math. Soc., 1957. 808 p.
  11. Engel K.-J., Nagel R. A short course on operator semigroups. New York, Universitext, Springer, 2006. 247 p.
  12. Daletsky Yu. L., Krein M. G. Stability of Solutions of Differential Equations in Banach Space. Moscow, Nauka, 1970. 535 p. (in Russian)
  13. Hardy G. H. A theorem concerning trigonometrical series. J. London Math. Soc., 1928, iss. 3, pp. 12–13.
  14. Seneta E. Regularly varying functions. Lecture Notes in Mathematics, vol. 508, Berlin, Heidelberg, New York, Springer-Verlag, 1976. 112 p. DOI: https://doi.org/10.1007/BFb0079658.
  15. Levin B. Ya. Distribution of zeros of entire functions. Moscow, Gostekhizdat, 1956. 632 p. (in Russian).
  16. Strukova I. I. Wiener’s theorem for periodic at infinity functions. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 4, pp. 34–41 (in Russian).
  17. Strukova I. I. About Wiener theorem for periodic at infinity functions. Siberian Math. J., 2016, vol. 57, no. 1, pp. 186–198 (in Russian).
  18. Strukova I. I. About harmonic analysis of periodic at infinity functions. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 28–38 (in Russian).
  19. Strukova I. I. Harmonic analysis of periodic vectors and functions periodic at infinity. J. Math. Sci., 2015, vol. 211, no. 6, pp. 874–885.
  20. Strukova I. I. Spectra of algebras of slowly varying and periodic at infinity functions and Banach limits. Proc. Voronezh State Univ. Ser. Physics. Mathematics, 2015, no. 3, pp. 161–165 (in Russian).
Received: 
11.01.2017
Accepted: 
29.04.2017
Published: 
31.05.2017