The basis property of the Legendre polynomials in the variable exponent Lebesgue space $ L^{p(x)}(-1,1)$
Идрис Идрисович Шарапудинов · Sbornik Mathematics · 2009
The paper looks at the problem of determining the conditions on a?variable exponent so that the orthonormal system of Legendre polynomials is a?basis in the Lebesgue space with norm 0:\int_{-1}^1\biggl\vert{\frac{f(x)}{\alpha}}\biggr\vert^{p(x)}\,dx \le1\biggr\}.$ SRC=http://ej.iop.org/images/1064-5616/200/1/A05/tex_sm_3989_img4.gif/> Conditions on the exponent , that are definitive in a?certain sense, are obtained and guarantee that the system has the basis property in . Bibliography: 31 titles.