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 Daghestan Electronic Mathematical Reports: Year: Volume: Issue: Page: Find

 Daghestan Electronic Mathematical Reports, 2017, Issue 8, Pages 27–47 (Mi demr46)

Convergence of Fourier series in Jacobi polynomials in weighted Lebesgue space with variable exponent

I. I. Sharapudinovab, T. N. Shakh-Emirova

a Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala
b Daghestan State Pedagogical University

Abstract: The problem of basis property of the Jacobi polynomials system $P_n^{\alpha,\beta}(x)$ in the weighted Lebesgue space $L^{p(x)}_\mu([-1,1])$ with variable exponent $p(x)$ and $\mu(x) = (1-x)^\alpha(1+x)^\beta$ is considered. It is shown that if $\alpha,\beta>-1/2$ and $p(x)$ satisfies on $[-1,1]$ some natural conditions then the orthonormal Jacobi polynomials system $p_n^{\alpha,\beta}(x)=(h_n^{\alpha,\beta})^{-\frac12}P_n^{\alpha,\beta}(x)$ $(n=0,1,\ldots)$ is a basis of $L^{p(x)}_\mu([-1,1])$ as $4\frac{\alpha+1}{2\alpha+3}<p(1)<4\frac{\alpha+1}{2\alpha+1}$, $4\frac{\beta+1}{2\beta+3}<p(-1)<4\frac{\beta+1}{2\beta+1}$.

Keywords: basis property of the Jacobi polynomials, Fourier-Jacobi sums, convergence in the weighted Lebesgue space with variable exponent, Dini-Lipshits condition

DOI: https://doi.org/10.31029/demr.8.4

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UDC: 517.538
Revised: 23.11.2017
Accepted:28.11.2017

Citation: I. I. Sharapudinov, T. N. Shakh-Emirov, “Convergence of Fourier series in Jacobi polynomials in weighted Lebesgue space with variable exponent”, Daghestan Electronic Mathematical Reports, 2017, no. 8, 27–47

Citation in format AMSBIB
\Bibitem{ShaSha17} \by I.~I.~Sharapudinov, T.~N.~Shakh-Emirov \paper Convergence of Fourier series in Jacobi polynomials in weighted Lebesgue space with variable exponent \jour Daghestan Electronic Mathematical Reports \yr 2017 \issue 8 \pages 27--47 \mathnet{http://mi.mathnet.ru/demr46} \crossref{https://doi.org/10.31029/demr.8.4}