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Mat. Zametki, 2019, Volume 106, Issue 4, Pages 595–621 (Mi mz11707)  

The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent

I. I. Sharapudinovabc

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

Abstract: The problem of the basis property of ultraspherical Jacobi polynomials in a Lebesgue space with variable exponent is studied. We obtain sufficient conditions on the variable exponent $p(x)>1$ that guarantee the uniform boundedness of the sequence $S_n^{\alpha,\alpha}(f)$, $n=0,1,…$, of Fourier sums with respect to the ultraspherical Jacobi polynomials $P_k^{\alpha,\alpha}(x)$ in the weighted Lebesgue space $L_\mu^{p(x)}([-1,1])$ with weight $\mu=\mu(x)=(1-x^2)^\alpha$, where $\alpha>-1/2$. The case $\alpha=-1/2$ is studied separately. It is shown that, for the uniform boundedness of the sequence $S_n^{-1/2,-1/2}(f)$, $n=0,1,…$, of Fourier–Chebyshev sums in the space $L_\mu^{p(x)}([-1,1])$ with $\mu(x)=(1-x^2)^{-1/2}$, it suffices and, in a certain sense, necessary that the variable exponent $p$ satisfy the Dini–Lipschitz condition of the form
$$ |p(x)-p(y)|\le \frac{d}{-\ln|x-y|}\mspace{2mu}, \qquadwhere\quad |x-y|\le \frac{1}{2},\quad x,y\in[-1,1],\quad d>0, $$
and the condition $p(x)>1$ for all $x\in[-1,1]$.

Keywords: the basis property of ultraspherical polynomials, Fourier–Jacobi sums, Fourier–Chebyshev sums, convergence in a weighted Lebesgue space with variable exponent, Dini–Lipschitz condition.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00486
This work was supported by the Russian Foundation for Basic Research under grant 16-01-00486.


DOI: https://doi.org/10.4213/mzm11707

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English version:
Mathematical Notes, 2019, 106:4, 616–638

Bibliographic databases:

UDC: 517.538
Received: 12.10.2018

Citation: I. I. Sharapudinov, “The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent”, Mat. Zametki, 106:4 (2019), 595–621; Math. Notes, 106:4 (2019), 616–638

Citation in format AMSBIB
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\by I.~I.~Sharapudinov
\paper The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent
\jour Mat. Zametki
\yr 2019
\vol 106
\issue 4
\pages 595--621
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\pages 616--638
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