Skip to main content
Log in

Spectral synthesis for the differentiation operator in the Schwartz space

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We consider the spectral synthesis problem for the differentiation operator on the space of infinitely differentiable functions on a finite or infinite interval of the real line and the dual problem of local description of closed submodules in a special module of entire functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. L. Euler, “De integratione aequationum differentialum altiorum gradum,” Miscellanea Berol., No. 7, 193–242 (1743).

    Google Scholar 

  2. L. Schwartz, “Théorie générale des fonctions moyenne-périodique,” Ann. of Math. (2) 48 (4), 857–929 (1947).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Aleman and B. Korenblum, “Derivation-invariant subspaces of C ,” Comput. Methods Funct. Theory 8 (2), 493–512 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Aleman, A. Baranov, and Yu. Belov, “Subspaces of C invariant under the differentiation,” J. Funct. Anal. 268 (8), 2421–2439 (2015).

    Article  MATH  MathSciNet  Google Scholar 

  5. N. F. Abuzyarova, “Spectral synthesis in a Schwartz space of infinitely differentiable functions,” Dokl. Ross. Akad. Nauk 457 (5), 510–513 (2014) [Dokl. Math. 90 (1), 479–482 (2014)].

    MATH  MathSciNet  Google Scholar 

  6. A. Beurling and P. Malliavin, “On the closure of characters and the zeros of entire functions,” Acta Math. 118 (1-4), 79–93 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  7. N. F. Abuzyarova, “Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis,” Ufimsk. Mat. Zh. 8 (1), 3–14 (2016) [Ufa Math. Journal 8 (1), 1–12 (2016)].

    Article  MathSciNet  Google Scholar 

  8. I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. I. Spectral analysis on convex regions,” Mat. Sb. 87 (129) (4), 459–489 (1972) [Math. USSR-Sb. 16 (4), 471–500 (1972)].

    MathSciNet  Google Scholar 

  9. L. Ehrenpreis, “Mean periodic functions. I. Varieties whose annihilator ideals are principal,” Amer. J. Math. 77 (2), 293–326 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Sebastia˜ o e Silva, “Su certe classi di spazi localmente convessi importanti per le applicazioni,” Rend. Mat. e Appl. (5) 14, 388–410 (1955).

    MATH  MathSciNet  Google Scholar 

  11. L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1 (Springer, Berlin, 1983).

    MATH  Google Scholar 

  12. N. F. Abuzyarova, “Closed submodules in the module of entire functions of exponential type and polynomial growth on the real axis,” Ufimsk. Mat. Zh. 6 (4), 3–18 (2014) [Ufa Math. Journal 6 (4), 3–17 (2014)].

    Article  MathSciNet  Google Scholar 

  13. B. Y. Levin, Lectures on Entire Functions, in Transl. Math. Monogr. (Amer. Math. Soc., Providence, RI, 1996), Vol. 150.

    Google Scholar 

  14. I. F. Krasichkov-Ternovskii, “Local description of closed ideals and submodules of analytic functions of one variable. I,” Izv. Akad. Nauk SSSR Ser. Mat. 43 (1), 44–66 (1979) [Math. USSR-Izv. 14 (1), 41–60 (1980)].

    MathSciNet  Google Scholar 

  15. I. F. Krasichkov-Ternovskii, “Local description of closed ideals and submodules of analytic functions of one variable. II,” Izv. Akad. Nauk SSSR Ser. Mat. 43 (2), 309–341 (1979) [Math. USSR-Izv. 14 (2), 289–316 (1980)].

    MathSciNet  Google Scholar 

  16. I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. III. On the extension of spectral synthesis,” Mat. Sb. 88 (130) (3 (7)), 331–352 (1972) [Math. USSR-Sb. 17) (3), 327–348 (1972)].

    MathSciNet  Google Scholar 

  17. C. A. Berenstein and B. A. Taylor, “A new look at interpolation theory for entire functions of one variable,” Adv. in Math. 33 (2), 109–143 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Koosis, The Logarithmic Integral. I, in Cambridge Stud. Adv. Math. (Cambridge Univ. Press, Cambridge, 1998), Vol. 12.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. F. Abuzyarova.

Additional information

Original Russian Text © N. F. Abuzyarova, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 2, pp. 163–177.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abuzyarova, N.F. Spectral synthesis for the differentiation operator in the Schwartz space. Math Notes 102, 137–148 (2017). https://doi.org/10.1134/S0001434617070161

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434617070161

Navigation