The basis property of the Legendre polynomials in the variable

I. I. Sharapudinov

Sbornik. Mathematics, Tome 200 (2009) no. 1, pp. 133-156

Résumé

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.

Détail

Keywords: variable exponent, basis.
Mots-clés : Lebesgue space, Legendre polynomial

Bibliographie

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