Convergence of gaussian quadrature formulas for power orthogonal polynomials

Abstract

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I = (a, b), a function G ∈ S(w):= {f: Σ _I|f(x)|w(x)dx < ∞} satisfying the conditions G ^(2j)(x) ≥ 0, x ∈ (a, b), j = 0, 1, …, and growing as fast as possible as x → a+ and x → b−, plays an important role. But to find such a function G is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying \mathop {\sup }\limits_n \sum\limits_{k = 1}^n {\lambda _{0kn} G(x_{kn} ) < \infty } instead, where the x _kn’s are the zeros of the nth power orthogonal polynomial with respect to the weight w and λ _0kn’s are the corresponding Cotes numbers. Furthermore, some results of the convergence for Gaussian quadrature formulas involving the above condition are given.

Keywords

Convergence / Gaussian quadrature formula / Freud weight

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