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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On Stieltjes polynomials and Gauss-Kronrod quadrature


Author: Franz Peherstorfer
Journal: Math. Comp. 55 (1990), 649-664
MSC: Primary 65D32; Secondary 33C45, 41A55
MathSciNet review: 1035940
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Abstract: Let D be a real function such that $ D(z)$ is analytic and $ D(z) \ne 0$ for $ \vert z\vert \leq 1$. Furthermore, put $ W(x) = \sqrt {1 - {x^2}} \vert D({e^{i\varphi }}){\vert^2}$, $ x = \cos \varphi $, $ \varphi \in [0,\pi ]$, and denote by $ {p_n}( \bullet ,W)$ the polynomial which is orthogonal on $ [ - 1, + 1]$ to $ {\mathbb{P}_{n - 1}}$ ( $ {\mathbb{P}_{n - 1}}$ denotes the set of polynomials of degree at most $ n - 1$) with respect to W. In this paper it is shown that for each sufficiently large n the polynomial $ {E_{n + 1}}( \bullet ,W)$ (called the Stieltjes polynomial) of degree $ n + 1$ which is orthogonal on $ [ - 1, + 1]$ to $ {\mathbb{P}_n}$ with respect to the sign-changing function $ {p_n}( \bullet ,W)W$ has $ n + 1$ simple zeros in $ ( - 1,1)$ and that the interpolation quadrature formula (called the Gauss-Kronrod quadrature formula) based on nodes which are the $ 2n + 1$ zeros of $ {E_{n + 1}}( \bullet ,W){p_n}( \bullet ,W)$ has all weights positive.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1990-1035940-8
PII: S 0025-5718(1990)1035940-8
Keywords: Stieltjes polynomials, orthogonal polynomials, Gauss-Kronrod quadrature formulas
Article copyright: © Copyright 1990 American Mathematical Society