Construction of σ-orthogonal polynomials and gaussian quadrature formulas

Abstract

Let dα be a measure on R and let σ = (m ₁, m ₂,...,m _n ), where m _k ≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ω _n (x) := (x − x ₁)(x − x ₂)...(x − x _n ) with x ₁ ≤ x ₂ ≤ ... ≤ x _n is said to be the nth σ-orthogonal polynomial with respect to dα if the vector of zeros (x ₁, x ₂, ..., x _n)^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 m _k 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 m _k are positive integers no less than 2 is provided. Finally, some numerical examples are also given.