Orthogonal Laurent Polynomials and Quadrature Formulas for Unbounded Intervals: I. Gauss-Type Formulas

Bultheel, Adhemar; Díaz-Mendoza, C.; González-Vera, Pablo; Orive, R.

doi:10.1216/rmjm/1181069968

Cited by 9 publications

(15 citation statements)

References 17 publications

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“…(see [7]), i.e., the corresponding quadrature rule exactly "integrates" any L-polynomial in L n−1 . On the other hand, let L n (f ; x) be the unique L-polynomial in L n−1 interpolating f at the n distinct nodes {x j } n j=1 , so that one can write…”

Section: Quadraturesmentioning

confidence: 99%

“…The quadrature formula I n (f ) given in Corollary 4.2 will be called L-orthogonal (see [7]) for the SSD φ and the "ordering" induced by {p(n)} n≥0 .…”

Section: Proof It Only Remains To Prove Thatmentioning

confidence: 99%

“…Further results on convergence of the L-orthogonal formula can be deduced from [6] and [7]. Concerning convergence of 2PA's see [16,27].…”

Section: Two-point Padé Approximationmentioning

confidence: 99%

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Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

2005

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Abstract. Starting from a strong Stieltjes distribution φ, general sequences of orthogonal Laurent polynomials are introduced and some of their most relevant algebraic properties are studied. From this perspective, the connection between certain quadrature formulas associated with the distribution φ and two-point Padé approximants to the Stieltjes transform of φ is revisited. Finally, illustrative numerical examples are discussed.

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Section: Quadraturesmentioning

confidence: 99%

“…The quadrature formula I n (f ) given in Corollary 4.2 will be called L-orthogonal (see [7]) for the SSD φ and the "ordering" induced by {p(n)} n≥0 .…”

Section: Proof It Only Remains To Prove Thatmentioning

confidence: 99%

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Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

2005

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“…,n, then b a f (t) dψ(t) = n i=1 λ n,i f (z n,i ) for t n f (t) ∈ P 2n−1 . (1.5) This is the quadrature rule of highest algebraic degree of precision associated with the L-orthogonal polynomial Q n (see, for example, [2] and [15]), which is analogous to the Gaussian rule associated with the n-th degree orthogonal polynomial on the real line.…”

Section: Introductionmentioning

confidence: 99%

Kernel polynomials from L-orthogonal polynomials

Félix

Ranga

Veronese

2011

Applied Numerical Mathematics

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A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0, ±1, ±2, . . . , is called a strong positive measure on [a, b]. If 0 a < b ∞ then the sequence of (monic) polynomials {Q n }, defined by b a t −n+s Q n (t) dψ(t) = 0, s = 0, 1, . . . ,n − 1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these Lorthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered.The numbers σ n,s = b a t −n+s Q n (t) dψ(t), which by (1.1) must satisfy σ n,s = 0 for 0 s n − 1, n 1, also satisfy ✩ This research was supported by grants from CAPES, CNPq and FAPESP of Brazil.

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“…[37, 38, 46-49, 61, 62, 66-73, 81-83, 87, 88, 96-99, 102-105, 108, 109]. Their investigations were deepened further in the subsequent period [117-119, 126-128, 132, 138-140, 142, 145, 154, 164, 173-175, 179, 183].With his PhD students he started several parallel research topics like Laurent Padé and Padé-type approximation (M. Camacho, PhD 1991) [51,52,74,100,101], multipoint Padé-type approximation and quadrature (M. Jiménez Paíz, PhD 1991) [15, 27, 30, 32, 41, 64, 80, 90, 112, 115], multivariate Padé approximation (R. Orive, PhD 1991), [42,59,60] composite and alternative numerical quadrature formulas (J.C. Santos León, PhD 1995) [19,29,43,44,57,63, 65,85,[91][92][93], Convergence of twopoint Padé-type approximants (C. Díaz Mendoza, PhD 2000) [89,94,107,114,116,130,137,144,149,151,152,170], quadrature on the unit circle (L. Daruis, PhD 2001) [113, 121-123, 125, 129, 134-136, 143, 146, 156, 158, 159, 167] which resulted in their book Ortogonalidad y Cuadraturas sobre la Circunferencia Unidad [153], orthogonal Laurent polynomials and quadrature on the unit circle and on the real line (R. …”

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confidence: 99%

Foreword to the proceedings of the OrthoQuad 2014 conference

Bultheel

Lagomasino

Marcellán

et al. 2015

Journal of Computational and Applied Mathematics

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OrthoQuad 2014 was an international symposium on orthogonality, quadrature, and related topics, held on January 20-24, 2014 in Puerto de la Cruz, Tenerife, Spain. It was held in memory of Professor Pablo González-Vera (1955-2012. A short CV of Pablo González-VeraPablo González-Vera was born in Vallehermoso (La Gomera, Canary Islands) on January 25, 1955. He studied Mathematics at University of La Laguna (1974-79), and began teaching at this university in 1980 and became Cátedratico (Full Professor) de Matemática Aplicada in 1992.He started his research in rational interpolation and Padé approximation, in collaboration with Professor Luis Casasús (now in Universidad Politécnica de Madrid) [1,3]. In particular Padé-type and two-point Padé approximation got his specific interest [4-7, 9-14, 20].This resulted in a Doctoral Dissertation on Two-point Padé Approximation [8] that he defended in 1985 at the University of La Laguna. His supervisor was Professor Nácere E. Hayek Calil.But soon orthogonality of the denominator polynomials of the approximants and Laurent polynomials got his interest and these showed up in another application, which became his most favored topic: numerical quadrature [2,[15][16][17][18][19][20].That was the germ of a research group on Approximation Theory in University of La Laguna. From 1991 on, eight doctoral thesis on Orthogonality and/or Quadrature were supervised by González-Vera, with the most recent being defended shortly after his passing away.Since 1989 a collaboration of the "gang of four" started: Adhemar Bultheel (KU Leuven, Belgium), Erik Hendriksen (University of Amsterdam, The Netherlands) and Olav Njåstad (University of Trondheim, Norway). In 1990 a local report of 84 pages [21] was published that formed the embryo of what became ten years later their book Orthogonal Rational Functions (Cambridge University Press, 1999) [110].During these ten years they collaborated intensively on the properties of these rational functions that generalized the orthogonal polynomials and Laurent polynomials. The support of the orthogonality measure could be the unit circle or on the real line (the whole real half line, the half line or a finite interval). They appeared in multipoint versions of Padé approximation, in generalizations of classical moment problems, and in quadrature formulas on the real line or the unit circle to generalize Gauss and Szegő numerical integration methods. [37, 38, 46-49, 61, 62, 66-73, 81-83, 87, 88, 96-99, 102-105, 108, 109]. Their investigations were deepened further in the subsequent period [117-119, 126-128, 132, 138-140, 142, 145, 154, 164, 173-175, 179, 183].With his PhD students he started several parallel research topics like Laurent Padé and Padé-type approximation (M. Camacho, PhD 1991) [51,52,74,100,101], multipoint Padé-type approximation and quadrature (M. Jiménez Paíz, PhD 1991) [15, 27, 30, 32, 41, 64, 80, 90, 112, 115], multivariate Padé approximation (R. Orive, PhD 1991), [42,59,60] composite and alternative numerical quadrature formulas (J...

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Orthogonal Laurent Polynomials and Quadrature Formulas for Unbounded Intervals: I. Gauss-Type Formulas

Cited by 9 publications

References 17 publications

Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

Kernel polynomials from L-orthogonal polynomials

Foreword to the proceedings of the OrthoQuad 2014 conference

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