Riemann boundary value problem with degenerate coefficients and bases from the double system of exponents in a generalized Lebesgue spaces

Abstract

In this paper a double system of exponents with degenerate coefficients is considered.Basicity of this system is studied in a generalized Lebesgue space $L_{p\left( \cdot \right)} $ . Method of boundaryvalue problems of the theory of analytic functions is applied. In addition, a specific Riemannboundary value problem with degenerate coefficients is obtained. At first, this problem isstudied in the Hardy classes with variable summability. The obtained results are applied to thestudy of basicity of considered double system of exponents in   $L_{p\left( \cdot \right)} $, when the coefficients areassumed a degeneration at the ends of the segment  $[-\pi, \pi]$.