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Mat. Sb., 2009, Volume 200, Number 1, Pages 137–160 (Mi msb4877)  

This article is cited in 8 scientific papers (total in 8 papers)

The basis property of the Legendre polynomials in the variable exponent Lebesgue space $L^{p(x)}(-1,1)$

I. I. Sharapudinovab

a Daghestan Scientific Centre of the Russian Academy of Sciences
b Vladikavkaz Scientific Centre of the Russian Academy of Sciences

Abstract: The paper looks at the problem of determining the conditions on a variable exponent $p=p(x)$ so that the orthonormal system of Legendre polynomials $\{\widehat P_n(x)\}_{n=0}^\infty$ is a basis in the Lebesgue space $L^{p(x)}(-1,1)$ with norm
$$ \|f\|_{p( \cdot )}=\inf\{\alpha>0: \int_{-1}^1|{\frac{f(x)}{\alpha}}|^{p(x)} dx \le1\}. $$
Conditions on the exponent $p=p(x)$, that are definitive in a certain sense, are obtained and guarantee that the system $\{\widehat P_n(x)\}_{n=0}^\infty$ has the basis property in $L^{p(x)}(-1,1)$.
Bibliography: 31 titles.

Keywords: Lebesgue space, variable exponent, Legendre polynomial, basis.

DOI: https://doi.org/10.4213/sm4877

Full text: PDF file (624 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:1, 133–156

Bibliographic databases:

UDC: 517.518.34
MSC: Primary 33A45; Secondary 42C10, 46E30
Received: 17.03.2008 and 30.11.2008

Citation: I. I. Sharapudinov, “The basis property of the Legendre polynomials in the variable exponent Lebesgue space $L^{p(x)}(-1,1)$”, Mat. Sb., 200:1 (2009), 137–160; Sb. Math., 200:1 (2009), 133–156

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. I. Sharapudinov, T. I. Sharapudinov, “Mixed Series of Jacobi and Chebyshev Polynomials and Their Discretization”, Math. Notes, 88:1 (2010), 112–139  mathnet  crossref  crossref  mathscinet  isi  elib
    2. Muradov T.R., “On basicity of perturbed system of exponents in Lebesgue space with variable summability factor”, Dokl. Math., 85:2 (2012), 219–221  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Sharapudinov I.I., “Approksimativnye svoistva srednikh valle-pussena na klassakh tipa soboleva s peremennym pokazatelem”, Vestnik Dagestanskogo nauchnogo tsentra RAN, 2012, no. 45, 5–13  elib
    4. I. I. Sharapudinov, “Priblizhenie gladkikh funktsii v $L^{p(x)}_{2\pi}$ srednimi Valle-Pussena”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(1) (2013), 45–49  mathnet
    5. I. I. Sharapudinov, “Prostranstvo Soboleva s peremennym pokazatelem i priblizhenie algebro-trigonometricheskimi polinomami”, Vestnik Dagestanskogo nauchnogo tsentra RAN, 2014, no. 53, 5–21  elib
    6. V.M. Keselman, “On a criterion of conformal parabolicity of a Riemannian manifold”, Sb. Math, 206:3 (2015), 389  mathnet  crossref  mathscinet  zmath  scopus  scopus
    7. Israfilov D.M., Yirtici E., “Convolutions and Best Approximations in Variable Exponent Lebesgue Spaces”, Math. Rep., 18:4 (2016), 497–508  mathscinet  zmath  isi
    8. I. I. Sharapudinov, T. N. Shakh-Emirov, “Skhodimost ryadov Fure po polinomam Yakobi v vesovom prostranstve Lebega s peremennym pokazatelem”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 8, 27–47  mathnet  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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