On Weighted Generalized Gauss Quadratures for Müntz Systems

Abstract

A novel recurrence formula for moments with respect to Müntz-Legendre polynomials is proposed and applied to construct a numerical method for solving generalized Gauss quadratures with power function weight for Müntz systems. These quadrature rules exhibit several properties similar to the classical Gaussian quadratures for polynomial systems, including positive weights, rapid convergence, and others. They are applicable to a wide range of functions, including smooth functions and functions with endpoint singularities, commonly found in integral equations with singular kernels, complex analysis, potential theory, and other areas.