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The Method of Boundary Value Problems in the Study of the Basis Properties of Perturbed System of Exponents in Banach Function Spaces

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Abstract

This paper considers the method of Riemann boundary value problems of the theory of analytic functions to study basis properties of perturbed systems of exponents in rearrangement invariant Banach function spaces. This method is demonstrated by the example of a system of exponents with a linear phase, depending on a complex parameter. The study of the basis properties of this system has a deep history starting with the classical results of Paley–Wiener and Levinson. The basis properties of this system in Lebesgue spaces were finally established in the works of Sedletskii and Moiseev. As special cases of rearrangement invariant Banach function spaces (r.i.s. for short), we can mention Lebesgue, Orlicz, Lorentz, Lorentz–Orlicz, Marcinkiewicz, grand-Lebesgue and other classical and non-standard spaces. A subspace of r.i.s. is considered where continuous functions are dense and the conditions on phase parameter are found, depending on the Boyd indices, which are sufficient for basicity of the system of exponents for this subspace. In the special case, where the Boyd indices coincide with each other, these conditions become a basicity criterion. A new proof method, different from the previously known ones, is proposed. In particular, previously known results are obtained for some Banach function spaces.

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References

  1. Adams, D.R.: Morrey Spaces. Springer, Switzherland (2016)

    Google Scholar 

  2. Alili, V.G.: Banach Hardy classes and some properties. Lobachevskii J. Math. 43(2), 284–292 (2022)

    Article  MathSciNet  Google Scholar 

  3. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)

    Google Scholar 

  4. Bilalov, B.T.: Basicity of some exponential, sine and cosine systems. Dif. Uravn. 26(1), 10–16 (1990). (in Russian)

  5. Bilalov, B.T.: Basis properties of some systems of exponents, cosines and sines. Siberian Math. J. 45(2), 264–273 (2004)

    Article  MathSciNet  Google Scholar 

  6. Bilalov, B.T.: The basis property of a perturbed system of exponentials in Morrey-type spaces. Siberian Math. J. 60(2), 249–271 (2019)

    Article  MathSciNet  Google Scholar 

  7. Bilalov, B.T., Gasymov, T.B.: On bases for direct decomposition. Dokl. Akad. Nauk 467(4), 381–384 (2016)

    Google Scholar 

  8. Bilalov, B.T., Gasymov, T.B., Guliyeva, A.A.: On solvability of Riemann boundary value problem in Morrey–Hardy classes. Turk. J. Math. 40(50), 1085–1101 (2016)

    Article  MathSciNet  Google Scholar 

  9. Bilalov, B.T., Guliyeva, A.A.: On basicity of exponential systems in Morrey-type spaces. Int. J. Math. 25(6), 10 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bilalov, B.T., Guseynov, Z.G.: Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterr. J. Math 9(3), 487–498 (2012)

    Article  MathSciNet  Google Scholar 

  11. Bilalov, B.T., Huseynli, A.A., El-Shabrawy, S.R.: Basis properties of trigonometric systems in weighted Morrey spaces. Azerb. J. Math. 9(2), 200–226 (2019)

    MathSciNet  Google Scholar 

  12. Bilalov, B.T., Sadigova, S.R.: On solvability in the small of higher order elliptic equations in rearrangement invariant spaces. Siber. Math. J. 63(3), 425–437 (2022). (published in Sibirsk. Mat. Zh. 63(3), 516–530 (2022))

  13. Bilalov, B.T., Sadigova, S.R., Alili, V.G.: On solvability of Riemann problems in Banach Hardy spaces. Filomat 35(10), 3331–3352 (2021)

    Article  MathSciNet  Google Scholar 

  14. Bilalov, B.T., Guliyeva, A.E.: Banach Hardy spaces, Cauchy formula and Riesz theorem. Azerb. J. Math. 10(2), 157–174 (2020)

    MathSciNet  Google Scholar 

  15. Bilalov, B.T., Seyidova, FSh.: Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces. Turk. J. Math. 43, 1850–1866 (2019)

    Article  MathSciNet  Google Scholar 

  16. Boyd, D.W.: Spaces between a pair of reflexive Lebesgue spaces. Proc. Am. Math. Soc. 18, 215–219 (1967)

    Article  MathSciNet  Google Scholar 

  17. Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Springer, Basel (2013)

    Book  Google Scholar 

  18. Formica, M.R., Giova, R.: Boyd indices in generalized grand Lebesgue spaces and applications. Mediter. J. Math. 12(3), 987–995 (2015)

    Article  MathSciNet  Google Scholar 

  19. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. Nauka, Moscow (1966). (in Russian)

  20. Israfilov, D.M., Tozman, N.P.: Approximation in Morrey–Smirnov classes. Azerb. J. Math. 1(1), 99–113 (2011)

    MathSciNet  Google Scholar 

  21. Kadets, M.I.: The exact value of the Paley–Wiener constant. DAN SSSR 155(6), 1253–1254 (1964). (in Russian)

  22. Karlovich, A.Y.: Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. arXiv:Math/0305108 [math.FA] (2003)

  23. Karlovich, AYu.: Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights. J. Oper. Theory 47, 303–323 (2002)

    MathSciNet  Google Scholar 

  24. Karlovich, AYu.: On the essential norm of the Cauchy singular operator in weighted rearrangement-invariant spaces. Integral Equ. Oper. Theory 38, 28–50 (2000)

    Article  MathSciNet  Google Scholar 

  25. Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-Standard Function Spaces, 2: Variable Exponent Hölder, Morrey-Campanato and Grand Spaces. Springer, Berlin (2016)

    Book  Google Scholar 

  26. Krein, S.G., Petunin, J.I., Semenov, E.M.: Interpolation of linear operators, Moscow, Nauka (1978) (in Russian)

  27. Kufner, A., John, O., Fucik, S.: Function Spaces. Noordhoff, Leyden (1977)

    Google Scholar 

  28. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979)

    Book  Google Scholar 

  29. Moiseev, E.I.: On basicity of sine and cosine systems. DAN SSSR 275(4), 794–798 (1984). (in Russian)

  30. Moiseev, E.I.: On basicity of one sine system. Differ. Uravn. 23(1), 177–179 (1987). (in Russian)

  31. Moiseev, E.I.: On solution of Frankl problem in special domain. Differ. Uravn. 28(4), 682–692 (1992). (in Russian)

  32. Ponomarev, S.M.: On an eigenvalue problem. DAN SSSR 249(5), 1068–1070 (1979). (in Russian)

  33. Sedletski, A.M.: Biorthogonal expansions of functions in series of exponents on intervals of the real axis. Usp. Mat. Nauk 37(5(227)), 51–95 (1982). (in Russian)

  34. Sharapudinov, I.I.: On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces. Azerb. J. Math. 4(1), 55–72 (2014)

    MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the Azerbaijan Science Foundation (Grant No. AEF-MQM-QA-1-2021-4(41)-8/02/1-M-02).

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Authors and Affiliations

  1. The Ministry of Science and Education of the Republic of Azerbaijan, Institute of Mathematics and Mechanics, 9, V. Bahabzade Street, Baku, AZ1141, Azerbaijan

    B. T. Bilalov & S. R. Sadigova

  2. Department of Mathematics, Yildiz Technical University, Davutpasha Street, 34220, Istanbul, Turkey

    B. T. Bilalov

  3. Khazar University, 41 Mehseti Street, Baku, Azerbaijan

    S. R. Sadigova

  4. Baku Slavic University, Baku, Azerbaijan

    V. G. Alili

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  1. B. T. Bilalov
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  2. S. R. Sadigova
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  3. V. G. Alili
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Correspondence to B. T. Bilalov.

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Communicated by Feng Dai.

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Bilalov, B.T., Sadigova, S.R. & Alili, V.G. The Method of Boundary Value Problems in the Study of the Basis Properties of Perturbed System of Exponents in Banach Function Spaces. Comput. Methods Funct. Theory 24, 101–120 (2024). https://doi.org/10.1007/s40315-023-00488-2

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  • DOI: https://doi.org/10.1007/s40315-023-00488-2

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