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Mathematics of Computation

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Orthogonal polynomials for refinable linear functionals

Authors: Dirk Laurie and Johan de Villiers
Journal: Math. Comp. 75 (2006), 1891-1903
MSC (2000): Primary 65D30, 42C40; Secondary 42C05, 65D07
Published electronically: May 23, 2006
MathSciNet review: 2240640
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Abstract: A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires $ O(n^2)$ rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.

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Additional Information

Dirk Laurie
Affiliation: Department of Mathematics, University of Stellenbosch, South Africa

Johan de Villiers
Affiliation: Department of Mathematics, University of Stellenbosch, South Africa

Keywords: Gaussian quadrature, refinable, orthogonal polynomials
Received by editor(s): October 22, 2004
Received by editor(s) in revised form: May 3, 2005
Published electronically: May 23, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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