On the convergence of the means spectral expansions corresponding to the pseudo-differential operators for distributions from the Sobolev–Liouville classes

  • Onur Alp İlhanEmail author
  • Shakirbay G. Kasimov
  • Mahkambek M. Babaev
  • Danyal Soybaş


We study the convergence of means of spectral expansions corresponding to positive self-adjoint elliptic pseudo-differential operators for distributions from the Sobolev–Liouville class.


Spectral expansion Convergence of the means of spectral expansions Elliptic pseudo-differential operator Distribution Sobolev–Liouville spaces 

Mathematics Subject Classification

Primary 47G30 Secondary 58J26 


  1. 1.
    Taylor, M.: Pseudodifferential Operators. Princeton University, Princeton (1981)CrossRefzbMATHGoogle Scholar
  2. 2.
    ll’in, V.A.: Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator. Uspekhi Mat. Nauk 23(2), 61–120 (1968)Google Scholar
  3. 3.
    Il’in, V.A.: Conditions for the convergence of spectral decompositions that correspond to selfadjoint extensions of elliptic operators. Differents. Uravn. 9(1), 49–73 (1973)Google Scholar
  4. 4.
    Il’in, V.A.: Spectral Theory of Differential Operators. Springer, Moscow (1991)Google Scholar
  5. 5.
    Alimov, Sh.A.: On spectral decompositions of functions in \(H_p^\alpha \). Mat. Sb. (N.S.) 101 (1), 3–20 (Russian)Google Scholar
  6. 6.
    Nikol’skii, S.M.: Approximation of Functions of Several Variables and Embedding Theorems, Izdat. Nauka, Moscow (1977) (in Russian)Google Scholar
  7. 7.
    Lions, J.L., Magenes, E.: Probl’emes aux limtes non homoge’nes et applications. Dunod, Paris (1968)Google Scholar
  8. 8.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)Google Scholar
  9. 9.
    Kasimov, S.G.: On the uniform convergence of spectral expansions for elliptic pseudodifferential operators. Uzb. Mat. Zh. 3, 18–24 (1991)MathSciNetGoogle Scholar
  10. 10.
    Kasimov, S.G.: Uniform convergence of spectral expansions corresponding to elliptic pseudodifferential operators of continuous functions in the Liouville classes. Differentsial’nye Uravneniya 32(10), 1371–1375 (1996)MathSciNetGoogle Scholar
  11. 11.
    Hörmander, L.: Linear Partial Differential Equations. Springer, New York (1963)Google Scholar
  12. 12.
    Alimov, S.A.: On the spectral decompositions of distributions. Doklady 331(6), 661–662 (1993)zbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Faculty of EducationErciyes UniversityMelikgazi/KayseriTurkey
  2. 2.Mathematics FacultyNational University of UzbekistanTashkentUzbekistan

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