On Stieltjes polynomials and Gauss-Kronrod quadrature
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Abstract:
Let D be a real function such that $D(z)$ is analytic and $D(z) \ne 0$ for $|z| \leq 1$. Furthermore, put $W(x) = \sqrt {1 - {x^2}} |D({e^{i\varphi }}){|^2}$, $x = \cos \varphi$, $\varphi \in [0,\pi ]$, 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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 649-664
- MSC: Primary 65D32; Secondary 33C45, 41A55
- DOI: https://doi.org/10.1090/S0025-5718-1990-1035940-8
- MathSciNet review: 1035940