Orthogonality of the Jacobi polynomials with negative integer parameters

Alfaro, M.; Morales, María Álvarez de; Rezola, M. L.

doi:10.1016/s0377-0427(01)00589-1

Cited by 27 publications

(33 citation statements)

References 4 publications

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“…The novelty of this paper consists of the interpolation type and approximation type results that connect Sobolev orthogonality with Best Approximation Theory. In a second paper, in 2000 (see [8]), the same authors generalized previous results by considering the Hermite interpolation scheme in the discrete part of the bilinear form (1), in such a way that with their conclusions, all the orthogonality results mentioned in this historical introduction can be derived (as also the one obtained two years later in [2], involving Jacobi polynomials with negative integer parameters; by the way, the same problem, in a different setting and with a different approach is considered in [12]). …”

Section: Introductionmentioning

confidence: 90%

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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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Section: Introductionmentioning

confidence: 90%

“…m } ∞ m=n is an OPS with respect to some regular linear functional u, then there exists a symmetric and quasi-definite real matrix A of order n, such that {Q m } ∞ m=0 is the MOPS associated with the Sobolev bilinear form defined by (2).…”

Section: Sobolev Orthogonal Polynomials and Interpolationmentioning

confidence: 99%

Linear interpolation and Sobolev orthogonality

Moreno

García−Caballero

2009

Journal of Approximation Theory

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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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“…In the case of d = 1, the equation (1.1) becomes the ordinary differential equation satisfied by the Gegenbauer polynomials. In this case, the problem of negative indices has been studied by several authors; we refer to [1,2,3,6] and the references therein. For d = 2, equation (1.1) is classical and can be traced back to Hermite.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials and partial differential equations on the unit ball

Piñar¹,

Xu²

2009

Proc. Amer. Math. Soc.

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Abstract. Orthogonal polynomials of degree n with respect to the weight function W µ (x) = (1 − x 2 ) µ on the unit ball in R are known to satisfy the partial differential equationThe singular case of µ = −1, −2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2, −3, . . . is shown to have complete polynomial solutions if the dimension d is odd. The orthogonality of the solution is also discussed.

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Orthogonality of the Jacobi polynomials with negative integer parameters

Cited by 27 publications

References 4 publications

Linear interpolation and Sobolev orthogonality

Linear interpolation and Sobolev orthogonality

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Orthogonal polynomials and partial differential equations on the unit ball

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