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On Wiener’s Theorem for functions periodic at infinity

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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.

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References

  1. Baskakov A. G. and Kaluzhina N. S., “Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations,” Math. Notes, 92, No. 5, 587–605 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  2. Baskakov A. G., “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations,” Russian Math. Surveys, 68, No. 1, 69–116 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  3. Baskakov A. G., Kaluzhina N. S., and Polyakov D. M., “Operator semigroups slowly varying at infinity,” Russian Math. (Iz. VUZ), 58, No. 7, 1–10 (2014).

    MathSciNet  MATH  Google Scholar 

  4. Baskakov A. G., “Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces,” Math. Notes, 97, No. 2, 164–178 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  5. Strukova I. I., “Wiener’s theorem for periodic functions at infinity,” Izv. Saratov. Univ. New Series. Math. Mech. Inform., 12, No. 4, 34–41 (2012).

    MATH  Google Scholar 

  6. Daletskiĭ Yu. L. and Kreĭn M. G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).

    Google Scholar 

  7. Pak I. N., “On the sums of trigonometric series,” Uspekhi Mat. Nauk, 35, No. 2, 91–144 (1980).

    MathSciNet  MATH  Google Scholar 

  8. Karamata M., “Sur un mode de croissance régulière. Théorèmes fondamentaux,” Bull. Soc. Math. France, No. 61, 55–62 (1933).

    MathSciNet  MATH  Google Scholar 

  9. Strukova I. I., “Harmonic analysis of periodic vectors and periodic functions at infinity,” Vestnik NGU. Ser. Mat. Mekh. Inform., 14, No. 1, 98–111 (2014).

    MATH  Google Scholar 

  10. Strukova I. I., “About harmonic analysis of periodic functions at infinity,” Izv. Saratov. Univ. New Series. Math. Mech. Inform., 14, No. 1, 28–38 (2014).

    MATH  Google Scholar 

  11. Wiener N., “Tauberian theorems,” Ann. Math., 33, No. 1, 1–100 (1932).

    Article  MathSciNet  MATH  Google Scholar 

  12. Strukova I. I., “Wiener’s theorem for periodic at infinity functions with summable weighted Fourier series,” Ufa Math. J., 5, No. 3, 140–148 (2013).

    Article  MathSciNet  Google Scholar 

  13. Baskakov A. G., “Spectral criteria for almost periodicity of solutions of functional equations,” Math. Notes, 24, No. 2, 606–612 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  14. Baskakov A. G., “Inequalities of Bernstein type in abstract harmonic analysis,” Sib. Math. J., 20, No. 5, 665–672 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  15. Baskakov A. G., “Spectral synthesis in Banach modules over commutative Banach algebras,” Math. Notes, 34, No. 4, 776–782 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  16. Baskakov A. G., “Harmonic analysis of cosine and exponential operator-valued functions,” Math. USSR-Sb., 52, No. 1, 63–90 (1985).

    Article  MATH  Google Scholar 

  17. Baskakov A. G., “Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators,” J. Math. Sci., 137, No. 4, 4885–5036 (2006).

    Article  MathSciNet  Google Scholar 

  18. Baskakov A. G. and Krishtal I. A., “Harmonic analysis of causal operators and their spectral properties,” Izv. Math., 69, No. 3, 439–486 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. Baskakov A. G. and Krishtal I. A., “Memory estimation of inverse operators,” J. Funct. Anal., 267, 2551–2605 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. Hewitt E. and Ross K., Abstract Harmonic Analysis. Vol. 2, Springer-Verlag, New York (1994).

  21. Baskakov A. G., “Abstract harmonic analysis and asymptotic estimates of elements of inverse matrices,” Math. Notes, 52, No. 2, 764–771 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  22. Baskakov A. G., “Estimates for the elements of inverse matrices and the spectral analysis of linear operators,” Izv. Math., 61, No. 6, 1113–1135 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  23. Balan R. and Krishtal I., “An almost periodic noncommutative Wiener’s lemma,” J. Math. Anal. Appl., 370, No. 2, 339–349 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  24. Baskakov A. G., “Asymptotic estimates for the entries of the matrices of inverse operators and harmonic analysis,” Sib. Math. J., 38, No. 1, 10–22 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  25. Gelfand I. M., Raĭkov D. A., and Shilov G. E., Commutative Normed Rings, Chelsea Publishing Company, New York (1964).

    Google Scholar 

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Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    I. I. Strukova

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  1. I. I. Strukova
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Correspondence to I. I. Strukova.

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Voronezh. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 1, pp. 186–198, January–February, 2016; DOI: 10.17377/smzh.2016.57.114.

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Strukova, I.I. On Wiener’s Theorem for functions periodic at infinity. Sib Math J 57, 145–154 (2016). https://doi.org/10.1134/S0037446616010146

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  • DOI: https://doi.org/10.1134/S0037446616010146

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