Gaussian-type Quadrature Rules for Müntz Systems

Abstract

A method for constructing the generalized Gaussian quadrature rules for Müntz polynomials on (0, 1) is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Müntz polynomials introduced by Badalyan [Akad. Nauk Armyan. SSR. Armenian summary)] and studied intensively by Borwein, Erdélyi, and Zhang [Trans. Amer. Math. Soc., 342 (1994), pp. 523-542], as well as by Milovanović on a numerical procedure for evaluation of such polynomials with high precision [Müntz orthogonal polynomials and their numerical evaluation, in

Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

“…This example is reported in [12], where a generalized Gaussian-type quadrature rule for Müntz system has been applied. Once the table of nodes and weights for a specific Müntz system is evaluated the algorithm of Milovanović and Cvetković [12] is more efficient than the one proposed in Section 4.2; however, it is practically impossible to list all the tables of nodes and weights for all kind of Müntz systems, and the computational cost of run-time generalized Gaussian quadrature is high.…”

“…In this paper we intend to construct an efficient quadrature algorithm for these polynomials and we observe that the polynomials presented here include and extend the functions presented in [9,12], where the generalized Gaussian quadrature is applied.…”

“…The algorithms of generalized Gaussian quadrature for nonclassical polynomials (in particular logarithmic functions or Müntz polynomials) are often ill conditioned, see [9], or in general need some effort to be implemented, see [12]. In particular, the algorithm of Milovanović and Cvetković [12] is stable and it is based on the high precision numerical evaluation of orthogonal Müntz polynomials to safely construct the quadrature rules.…”

“…In particular, the algorithm of Milovanović and Cvetković [12] is stable and it is based on the high precision numerical evaluation of orthogonal Müntz polynomials to safely construct the quadrature rules. However, it is practically impossible to list all the tables of nodes and weights for all kind of Müntz systems and the computational cost of run-time generalized Gaussian quadrature is high.…”

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

“…Problems with algebraic or/and logarithmic singularities can be solved using Müntz systems and quadratures of this type (cf. [30,32,11]) or using a procedure proposed in [20].…”