QUADRATURES WITH MULTIPLE NODES FOR FOURIER-CHEBYSHEV COEFFICIENTS

Abstract

Gaussian quadrature formulas, relative to the Chebyshev weight functions, with multiple nodes and their optimal extensions for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, are considered. The existence and uniqueness of such quadratures is proved. One of them is a generalization of the well-known Micchelli-Rivlin quadrature formula. The others are new. Numerically stable construction of these quadratures is proposed. By determining the absolute value of the difference between these Gaussian quadratures with multiple nodes for the Fourier-Chebyshev coefficients and their corresponding optimal extensions we get the well-known methods for estimation their error. Numerical results are included. These results are continuation of the recent ones by Bojanov and Petrova (J. Comput. Appl. Math., 2009), and Milovanovi´c and Spalevi´c (Math. Comp., 2014).