Geodesic
The basis property of the Legendre polynomials in the variable
Sbornik. Mathematics, Tome 200 (2009) no. 1, pp. 133-156

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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: variable exponent, basis.
Mots-clés : Lebesgue space, Legendre polynomial 
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     author = {I. I. Sharapudinov},
     title = {The basis property of the {Legendre} polynomials in the variable},
     journal = {Sbornik. Mathematics},
     pages = {133--156},
     publisher = {mathdoc},
     volume = {200},
     number = {1},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2009_200_1_a4/}
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I. I. Sharapudinov. The basis property of the Legendre polynomials in the variable. Sbornik. Mathematics, Tome 200 (2009) no. 1, pp. 133-156. http://geodesic.mathdoc.fr/item/SM_2009_200_1_a4/