Construction of σ-orthogonal polynomials and gaussian quadrature formulas
Tóm tắt
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.
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