Email this article The basis property of the Legendre polynomials in variable exponent Lebesgue space
- Authors: Magomed-Kasumov M.G.1,2, Shakh-Emirov T.N.1, Gadzhimirzaev R.M.1
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Affiliations:
- Daghestan Federal Research Centre of the Russian Academy of Sciences
- Vladikavkaz Scientific Centre of the Russian Academy of Sciences
- Issue: Vol 215, No 2 (2024)
- Pages: 103-119
- Section: Articles
- URL: https://ogarev-online.ru/0368-8666/article/view/251800
- DOI: https://doi.org/10.4213/sm9891
- ID: 251800
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Abstract
Sharapudinov proved that the Legendre polynomials form a basis of the Lebesgue space with variable exponent p(x) if p(x)>1 satisfies the Dini–Lipschitz condition and is constant near the endpoints of the orthogonality interval. We prove that the system of Legendre polynomials forms a basis of these spaces without the condition that the variable exponent be constant near the endpoints.
About the authors
Magomedrasul Grozbekovich Magomed-Kasumov
Daghestan Federal Research Centre of the Russian Academy of Sciences; Vladikavkaz Scientific Centre of the Russian Academy of Sciences
Author for correspondence.
Email: rasuldev@gmail.com
ORCID iD: 0000-0002-6624-5612
Candidate of physico-mathematical sciences, Senior Researcher
Tadgidin Nurmagomedovich Shakh-Emirov
Daghestan Federal Research Centre of the Russian Academy of Sciences
Email: Tadgius@gmail.com
Ramis Makhmudovich Gadzhimirzaev
Daghestan Federal Research Centre of the Russian Academy of Sciences
Email: ramis3004@gmail.com
ORCID iD: 0000-0002-6686-881X
Scopus Author ID: 57203401601
ResearcherId: AAE-1094-2022
Candidate of physico-mathematical sciences, Researcher
References
- H. Pollard, “The mean convergence of orthogonal series. I”, Trans. Amer. Math. Soc., 62:3 (1947), 387–403
- J. Newman, W. Rudin, “Mean convergence of orthogonal series”, Proc. Amer. Math. Soc., 3:2 (1952), 219–222
- И. И. Шарапудинов, “О базисности системы полиномов Лежандра в пространстве Лебега $L^{p(x)}(-1,1)$ с переменным показателем $p(x)$”, Матем. сб., 200:1 (2009), 137–160
- И. И. Шарапудинов, “О топологии пространства $mathscr L^{p(t)}([0,1])$”, Матем. заметки, 26:4 (1979), 613–632
- И. И. Шарапудинов, “О базисности системы Хаара в пространстве $mathscr L^{p(t)}([0,1])$ и принципе локализации в среднем”, Матем. сб., 130(172):2(6) (1986), 275–283
- D. V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces. Foundations and harmonic analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013, x+312 pp.
- Г. Сегe, Ортогональные многочлены, Физматгиз, М., 1962, 500 с.
- L. Diening, M. Růžička, “Calderon–Zygmund operators on generalized Lebesgue spaces $L^{p(cdot)}$ and problems related to fluid dynamics”, J. Reine Angew. Math., 2003:563 (2003), 197–220
- V. Kokilashvili, S. Samko, “Singular integrals in weighted Lebesgue spaces with variable exponent”, Georgian Math. J., 10:1 (2003), 145–156
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