Orthogonal polynomials and Gaussian quadrature for refinable weight functions

Laurie, Dirk; Villiers, J. M. De

doi:10.1016/j.acha.2004.06.002

Cited by 10 publications

(16 citation statements)

References 11 publications

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“…The need to evaluate φ is avoided by Laurie and De Villiers [11], generalizing an idea of Sweldens and Piessens [17]. This is that having chosen an auxiliary basis {π l , l = 0, 1, 2, .…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

“…We shall say nothing further about the interplay of properties of γ with those of φ, such as necessary and sufficient conditions for the existence, continuity, smoothness, etc., of φ when γ is given. There is a rich literature on that topic, some highlights of which are summarized in [11].…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

“…In [14] and [8] these properties are essential; in [17] and [11] they influence the choice of an auxiliary basis.…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

“…In [11], algorithms are derived which require only that the auxiliary basis be generated by a three-term recursion similar to (2.2)-(2.4)-no properties of the auxiliary basis beyond the numerical values of the recursion coefficients are required.…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

“…Three possibilities for the auxiliary basis are considered in [11]. Algorithm 1 uses the monomials (which trivially satisfy a three-term recursion with all coefficients zero), working in rational arithmetic because of the notorious ill-conditioning [6] of the map from the monomials to the orthogonal polynomials.…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

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Orthogonal polynomials for refinable linear functionals

Laurie

Villiers

2006

Math. Comp.

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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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“…The need to evaluate φ is avoided by Laurie and De Villiers [11], generalizing an idea of Sweldens and Piessens [17]. This is that having chosen an auxiliary basis {π l , l = 0, 1, 2, .…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

“…In [14] and [8] these properties are essential; in [17] and [11] they influence the choice of an auxiliary basis.…”

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

Section: Orthogonal Polynomials For Refinable Linear Functionals 1893mentioning

confidence: 99%

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Orthogonal polynomials for refinable linear functionals

Laurie

Villiers

2006

Math. Comp.

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Direct and inverse computation of Jacobi matrices of infinite iterated function systems

Mantica

2013

Numer. Math.

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We introduce a new set of algorithms to compute the Jacobi matrices associated with invariant measures of infinite iterated function systems, composed of one–dimensional, homogeneous affine maps. We demonstrate their utility in the study of theoretical problems, like the conjectured almost periodicity of such Jacobi matrices, the singularity of the measures, and the logarithmic capacity of their support. Since our technique is based on a reversible transformation between pairs of Jacobi matrices, it can also be applied to solve an inverse/approximation problem. The proposed algorithms are tested in significant, highly sensitive cases: they perform in a stable fashion, and can reliably compute Jacobi matrices of large order

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