Algebraic Methods for Modified Orthogonal Polynomials

Galant, David

doi:10.2307/2153072

Cited by 13 publications

(24 citation statements)

References 5 publications

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“…There are several contributions in such a direction; for instance, the works by Galant [7,8], Golub and Kautsky [10,15], and Gautschi [9]. Suppose that a given linear functional L is defined in terms of a weight function ω in the following way:…”

Section: Introductionmentioning

confidence: 99%

“…In the case of positive measures, there are many contributions (see [7][8][9]21]) when rational perturbationsω(x) = p(x) q(x) ω(x) of a weight function are considered. Grünbaum and Haine introduced a modification of such algorithms which allows to find the sequence of monic polynomials orthogonal with respect toL 3 in an alternative way to the approach by Maroni [17].…”

Section: Introductionmentioning

confidence: 99%

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Darboux transformation and perturbation of linear functionals

Bueno¹

2004

Linear Algebra and its Applications

106

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Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and 1 x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(x k ) = δ k,0 , and δ k,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with x 2 L as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Darboux transformation and perturbation of linear functionals

Bueno¹

2004

Linear Algebra and its Applications

106

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“…the integral of the modified weight), this result appears as the unnumbered theorem on p. 542 of [7].…”

Section: Conditioningmentioning

confidence: 82%

Computation of connection coefficients and measure modifications for orthogonal polynomials

Narayan

Hesthaven

2011

Bit Numer Math

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We observe that polynomial measure modifications for families of univariate orthogonal polynomials imply sparse connection coefficient relations. We therefore propose connecting L 2 expansion coefficients between a polynomial family and a modified family by a sparse transformation. Accuracy and conditioning of the connection and its inverse are explored. The connection and recurrence coefficients can simultaneously be obtained as the Cholesky decomposition of a matrix polynomial involving the Jacobi matrix; this property extends to continuous, non-polynomial measure modifications on finite intervals. We conclude with an example of a useful application to families of Jacobi polynomials with parameters (γ, δ ) where the fast Fourier transform may be applied in order to obtain expansion coefficients whenever 2γ and 2δ are odd integers.

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“…The problem of the numerical computation of the Geronimus transformation with shift α and parameter C of a Jacobi matrix J has been extensively studied when C = 0 and the shift α is close to the support of the measure μ [5,8,10]. However, we have not found any papers on the case C = 0, or when C = 0 and the shift is not close to the support of the measure.…”

Section: (X − α)G = L If and Only Ifmentioning

confidence: 99%

“…This application is connected to the problem of classifying all sequences of orthogonal polynomials such that its derivatives form another set of orthogonal polynomials. In the last two decades, these transformations have attracted the interest of various specialists in different branches of mathematics and mathematical physics for their applications to different topics such as Discrete Integrable Systems [20,22,23], Quantum Mechanics, Bispectral Transformations in Orthogonal Polynomials [16][17][18], and Numerical Analysis [5,7,8,10,12].…”

Section: (X − α)G = L If and Only Ifmentioning

confidence: 99%