Construction of σ-orthogonal polynomials and gaussian quadrature formulas

Abstract

Let dα be a measure on R and let σ = (m1, m2,...,mn), where mk ≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ωn(x) := (x − x1)(x − x2)...(x − xn) with x1 ≤ x2 ≤ ... ≤ xn is said to be the nth σ-orthogonal polynomial with respect to dα if the vector of zeros (x1, x2, ..., xn)T is a solution of the extremal problem $${\int_R {{\prod\limits_{k = 1}^n {{\left| {x - x_{k} } \right|}^{{m_{k} }} d\alpha {\left( x \right)} = {\mathop {\min }\limits_{y_{1} \leqslant y_{2} \leqslant ... \leqslant y_{n} } }} }} }\;{\int_R {{\prod\limits_{k = 1}^n {{\left| {x - y_{k} } \right|}^{{m_{k} }} d\alpha {\left( x \right)}.} }} }$$ In this paper the existence, uniqueness, characterizations, and continuity with respect to σ of a σ-orthogonal polynomial under a more mild assumption are established. An efficient iterative method based on solving the system of normal equations for constructing a σ-orthogonal polynomial is presented when all the mk are arbitrary real numbers no less than 2. A simple method to calculate the Cotes numbers of the corresponding generalized Gaussian quadrature formula when all the mk are positive integers no less than 2 is provided. Finally, some numerical examples are also given.