The basis property of the Legendre polynomials in the variable exponent Lebesgue space Lp(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 {P-circumflexn(x)}n=0∞ is a basis in the Lebesgue space Lp(x)(-1,1) with norm ||f||p(.)=inf{α>0:∫-11|f(x)/α|p(x)dx≤1}. Conditions on the exponent p=p(x), that are definitive in a certain sense, are obtained and guarantee that the system {P-circumflexn(x)}n=0∞ has the basis property in Lp(x)(-1,1). Bibliography: 31 titles.