The basis property of the Legendre polynomials in the variable exponent Lebesgue space $L^{p(x)}(-1,1)$

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 $L^{p(x)}(-1,1)$.
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