Sobolev orthogonal polynomials: The discrete-continuous case

Alfaro, M.; Pérez, Teresa E.; Piñar, Miguel A.; Rezola, M. L.

doi:10.4310/maa.1999.v6.n4.a10

Cited by 34 publications

(33 citation statements)

References 6 publications

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“…. , n − 1 the so-called fundamental polynomials l j = n−1 k=0 α k j e k belong to P n−1 and verify (3). Note that (l k ) n−1 k=0 = (e k ) n−1 k=0 G −1 .…”

Section: The Sobolev Discrete-continuous Bilinear Formmentioning

confidence: 93%

Linear interpolation and Sobolev orthogonality

Moreno

García−Caballero

2009

Journal of Approximation Theory

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There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints x * i , defined bywhere f belongs to a certain Sobolev space, a i j (·) are piecewise continuous functions over [a, b], b i jk are real numbers, and the points t k belong to [a, b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in P l , denoted by p f , which fulfills the interpolatory data x * i ( p f ) = x * i ( f ), and also the condition that p (n)f best approximates f (n) in P l−n (with respect to the norm induced by the continuous part of the original discrete-continuous bilinear form considered).

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“…. , n − 1 the so-called fundamental polynomials l j = n−1 k=0 α k j e k belong to P n−1 and verify (3). Note that (l k ) n−1 k=0 = (e k ) n−1 k=0 G −1 .…”

Section: The Sobolev Discrete-continuous Bilinear Formmentioning

confidence: 93%

Linear interpolation and Sobolev orthogonality

Moreno

García−Caballero

2009

Journal of Approximation Theory

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“…If we take γ + 1 = −N and δ → ∞ in the definition (19) of the Racah polynomials, we obtain the Hahn polynomials defined by (2). Hence…”

Section: Limit Relations Between Hypergeometric Orthogonal Polynomialsmentioning

confidence: 98%

“…In that case we obtain the Hahn polynomials given by (2) in the following way: lim δ→∞ R n (λ(x); −N − 1, β + γ + N + 1, γ , δ) = h γ ,β n (x; N).…”

Section: Limit Relations Between Hypergeometric Orthogonal Polynomialsmentioning

confidence: 98%

Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Costas-Santos

Sánchez-Lara

2009

Journal of Computational and Applied Mathematics

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MSC: 33C45 42C05 34B24Keywords: Classical orthogonal polynomials Inner product involving difference operators Non-standard orthogonality a b s t r a c t It is well-known that the family of Hahn polynomials {h α,β n (x; N)} n≥0 is orthogonal with respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for Hahn polynomials for a degree higher than N.We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials and give the limit relations among them for all n ∈ N 0 . Furthermore, in order to get these results for the Krawtchouk polynomials we will obtain a more general property of orthogonality for Meixner polynomials.

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“…In this respect, we refer the reader to [2][3][4]11,12] where general results on the Sobolev orthogonality of the Jacobi or Gegenbauer polynomials when one or both parameters α and β are negative integers. (…”

Section: Introductionmentioning

confidence: 99%

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Bruder

Littlejohn

2011

Results. Math.

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In this paper, we consider the second-order Jacobi differential expressionhere, the Jacobi parameters are α > −1 and β = −1. This is a nonclassical setting since the classical setting for this expression is generally considered when α, β > −1. In the classical setting, it is well-known that the Jacobi polynomials {P (α,β) n } ∞ n=0 are (orthogonal) eigenfunctions of a self-adjoint operator T α,β , generated by the Jacobi differential expression, in the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) β ). When α > −1 and β = −1, the Jacobi polynomial of degree 0 does not belong to the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) −1 ). However, in this paper, we show that the full sequence of Jacobi polynomials {P (α,−1) n } ∞ n=0 forms a complete orthogonal set in a Hilbert-Sobolev space Wα, generated by the inner productWe also construct a self-adjoint operator Tα, generated by α,−1 [·] in Wα, that has the Jacobi polynomials {P (α,−1) n } ∞ n=0 as eigenfunctions. Mathematics Subject Classification (2010). Primary 33C45, 34B30, 47B25; Secondary 34B20, 47B65. 284 A. Bruder and L. L. Littlejohn Results. Math.

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Sobolev orthogonal polynomials: The discrete-continuous case

Cited by 34 publications

References 6 publications

Linear interpolation and Sobolev orthogonality

Linear interpolation and Sobolev orthogonality

Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

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