Extensions of nonclosable operators and multiplication of distributions

  • Anatolij B. Antonevich Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Cemal Dolicanin State University of Novi Pazar, Vuka Karadzica, bb, Novi Pazar 36300, Serbia
DOI: https://doi.org/10.33581/2520-6508-2019-3-6-17

Abstract

In the paper some new constructions of extensions of nonclosable operators is proposed and several examples of applications are given. One of particular cases of the problem under consideration is the question on multiplication of distributions, a solution to which can be given by introduction of the so-called new generalized functions. It was demonstrated that the main obstacle for multiplication of distributions is nonclosablility of classical multiplication and the construction of new generalized functions is based on the ideas similar to that used under construction of the extension of nonclosable operators.

Author Biographies

Anatolij B. Antonevich, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of functional analysis and analytical economics, faculty of mechanics and mathematics

Cemal Dolicanin, State University of Novi Pazar, Vuka Karadzica, bb, Novi Pazar 36300, Serbia

doctor of science (mathematics); professor of mathematics

References

  1. Burachewski A, Radyno Ya, Antonevich A. On closability of nonclosable operators. Panamerican Mathematical Journal. 1997; 7(4):37–51.
  2. Schwartz L. Sur l’impossibilité de la multiplication des distributions. Comptes rendus de l’Académie des Sciences. 1954;239:847– 848.
  3. Ivanov VK. [Hyperdistributions and the multiplication of Schwartz distributions]. Doklady AN SSSR. 1972;204(5):1045–1048. Russian.
  4. Colombeau JF, editor. New generalized functions and multiplication of distributions. Amsterdam: North-Holland; 1984. 375 p. (North-Holland Mathematics Studies; volume 84).
  5. Rosinger EE. Generalized solutions of nonlinear partial differential equations. Amsterdam: Elsevier Science; 1987. 409 p. (North-Holland Mathematics Studies; volume 146).
  6. Egorov YuV. [A contribution to the theory of generalized functions]. Uspekhi matematicheskikh nauk. 1990;45(5):3– 40. Russian.
  7. Oberguggenberger M. Multiplication of distributions and application to partial differential equations (Pitman Research Notes in Mathematics). Harlow: Longman Higher Education; 1992. 336 p.
  8. Rosinger EE. Singularities and differential algebras of generalized functions: a basic dichotomic shealf theoretic singularity test. [S. l.]: Lambert Academic Publishing; 2013. 192 p.
  9. Kot MG. Asymptotics of the eigenvalues of approximating differential equations with d-different coefficients. Journal of the Belarusian State University. Mathematics and Informatics. 2017;1:4 –10. Russian.
  10. Shkadzinskaya AV. On equations containing derivative of the delta-function. Journal of the Belarusian State University. Mathematics and Informatics. 2017;3:19–26. Russian.
  11. Vladimirov VS. Obobshchennye funktsii v matematicheskoi fizike [Generalized functions in mathematical physics]. Moscow: Nauka; 1979. Russian.
  12. Dieudonne J. Foundations of modern analysis. New York: Academic Press; 1960. Russian edition: Dieudonne J. Osnovy sovremennogo analiza. Moscow: Mir; 1964.
  13. Mishchenko AS. Vektornye rassloeniya i ikh primeneniya [Vector bundles and their applications]. Moscow: Nauka; 1984. Russian.
  14. Bourbaki N. Theories Spectrales. Chapitres 1 et 2. Paris: Hermann; 1967. Russian edition: Burbaki N. Spektral’naya teoriya. Moscow: Mir; 1973.
  15. Antosik P, Mikusinski Ja, Sikorski R. Theory of distributions. The sequential approach. Amsterdam: Elsevier Scientific; 1973. Russian edition: Antosik P, Mikusinskii Ya, Sikorskii R. Teoriya obobshchennykh funktsii. Sekventsial’nyi podkhod. Moscow: Mir; 1976.
  16. Davis M. Applied nonstandard analysis. New York: Wiley; 1977. Russian edition: Devis M. Prikladnoi nestandartnyi analiz. Moscow: Mir; 1980.
Published
2019-11-26
Keywords: extension of operator, closed operator, multiplication of distributions
How to Cite
Antonevich, A. B., & Dolicanin, C. (2019). Extensions of nonclosable operators and multiplication of distributions. Journal of the Belarusian State University. Mathematics and Informatics, 3, 6-17. https://doi.org/10.33581/2520-6508-2019-3-6-17
Issue
No 3 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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