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Mathematics of Computation

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Linear combinations of orthogonal polynomials generating positive quadrature formulas

Author: Franz Peherstorfer
Journal: Math. Comp. 55 (1990), 231-241
MSC: Primary 65D32; Secondary 41A55, 42C05
MathSciNet review: 1023052
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Abstract: Let $ {p_k}(x) = {x^k} + \cdots $, $ k \in {{\mathbf{N}}_0}$, be the polynomials orthogonal on $ [ - 1, + 1]$ with respect to the positive measure $ d\sigma $. We give sufficient conditions on the real numbers $ {\mu _j}$, $ j = 0, \ldots ,m$, such that the linear combination of orthogonal polynomials $ \sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has n simple zeros in $ ( - 1, + 1)$ and that the interpolatory quadrature formula whose nodes are the zeros of $ \sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has positive weights.

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Keywords: Quadrature formula, positive weights, orthogonal polynomials, zeros
Article copyright: © Copyright 1990 American Mathematical Society