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This article is cited in 12 scientific papers (total in 12 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
$$
\|f\|_{p(\,\cdot\,)}=\inf\biggl\{\alpha>0:
\int_{-1}^1\biggl|{\frac{f(x)}{\alpha}}\biggr|^{p(x)}\,dx \le1\biggr\}.
$$
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.
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
Linking options:
https://www.mathnet.ru/eng/sm4877https://doi.org/10.1070/SM2009v200n01ABEH003989 https://www.mathnet.ru/eng/sm/v200/i1/p137
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Abstract page: | 851 | Russian version PDF: | 316 | English version PDF: | 39 | References: | 69 | First page: | 13 |
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