Acta Mathematica Hungarica

, Volume 157, Issue 1, pp 141–153 | Cite as

Spectral synthesis for the space of tempered solutions of a convolution system on discrete Abelian groups

  • S. S. PlatonovEmail author


We consider a problem of spectral synthesis in the topological vector space \({\mathcal{M}(G)}\) of tempered functions on a discrete Abelian group G. It is proved that the space of tempered solutions of a convolution system on discrete Abelian groups admits spectral synthesis, that is the space of tempered solutions of a convolution system coincides with the closed linear span in \({\mathcal{M}(G)}\) of all exponential monomial solutions of this system.

Key words and phrases

spectral synthesis convolution equation tempered functions on groups 

Mathematics Subject Classification

43A45 43A25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss., vol. 223, Springer-Verlag (Berlin–New York, 1976).Google Scholar
  2. 2.
    Bruhat, F.: Distributions sur un groupe localement compact et applications à l'étudedes représentations des groupes \(p\)-adiques. Bull. Soc. Math. France 89, 43–75 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gurevich, D.I.: Counterexamples to a problem of L. Schwartz. Funct. Anal. Appl. 9, 116–120 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Grundlehren Math. Wiss., vol. 115, Springer-Verlag (Berlin–Göttingen–Heidelberg, 1963)Google Scholar
  5. 5.
    Laczkovich, M., Székelyhidi, L.: Spectral synthesis on discrete groups. Math. Proc. Cambridge Philos. Soc. 143, 103–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Osborne, M.S.: On the Schwartz-Bruhat space and Paley-Wiener theorem for locally compact Abelian groups. J. Funct. Anal. 19, 40–49 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Platonov, S.S.: Spectral synthesis in some topological vector spaces of functions. St. Petersburg Math. J. 22, 813–833 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Platonov, S.S.: On spectral synthesis on zero-dimensional Abelian groups. Sb. Math. 204, 1332–1346 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Platonov, S.S.: On spectral synthesis on element-wise compact Abelian groups. Sb. Math. 206, 1150–1172 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. S. Platonov, On spectral analysis and spectral synthesis in the space of tempered functions on discrete abelian groups, J. Fourier Anal. Appl. (in press),
  11. 11.
    A. Robertson and W. Robertson, Topological Vector Spaces, Cambrige Tracts in Math., vol. 53, Cambrige Univ. Press (Cambrige, 1964)Google Scholar
  12. 12.
    Schwartz, L.: Théorie générale des fonctions moynne-périodiques. Ann. of Math. 48, 875–929 (1947)CrossRefGoogle Scholar
  13. 13.
    Schwartz, L.: Analyse et synthése harmonique dans les espaces de distributions. Canad. J. Math. 3, 503–512 (1951)CrossRefzbMATHGoogle Scholar
  14. 14.
    L. Székelyhidi, Discrete Spectral Synthesis and its Applications, Springer (Berlin, 2006)Google Scholar
  15. 15.
    Székelyhidi, L.: On the principal ideal theorem and spectral synthesis on discrete Abelian groups. Acta Math. Hungar. 150, 228–233 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Székelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientific Publishing Co. (Singapore etc., 1991)Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Institute of MathematicsPetrozavodsk State UniversityPetrozavodskRussia

Personalised recommendations