Some Additional Properties on Gaussian Quadrature Rules Obtained from Quasi-Symmetric Orthogonal Polynomials

Abstract

Recently, quasi-symmetric orthogonal polynomials and associated Gaussian quadrature rules were studied in [3]. It is well known that orthogonal polynomials on the real line satisfy a three-term recurrence relation (see [1]). In the quasi-symmetric case, the orthogonal polynomials satisfy the following three-term recurrence relation: P_n+1^φ(x) = (x − β_n+1^φ)P_n^φ(x) − γ_n+1^φP_n−1^φ(x), n ≥ 1, with P₀^φ(x) = 1, P₁^φ(x) = x − β₁^φ, β_2n−1^φ = a, β_2n^φ = b, a, b ∈ R, and max{|a|, |b|} > 0. In this work, for b > 0, and considering a = −b, we show that when f is an even function defined in I_a^b the weights in the quadrature rules can be simply replaced by the symmetric ones. Moreover, we also show that the error in the approximation is preserved. [...]