Basis property of the Legendre polynomials in variable exponent Lebesgue spaces

Sharapudinov proved that the Legendre polynomials form a basis of the Lebesgue space with variable exponent $p(x)$ if $p(x) > 1$ satisfies the Dini–Lipschitz condition and is constant near the endpoints of the orthogonality interval. We prove that the system of Legendre polynomials forms a basis of these spaces without the condition that the variable exponent be constant near the endpoints.
Bibliography: 9 titles.