Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Bruder, Andrea; Littlejohn, Lance L.

doi:10.1007/s00025-011-0102-4

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…Since the proofs of the next two results are similar to, respectively, the proofs given in Theorem 7 and Lemma 8 in [11], we omit them. We are now in position to prove the main result of this section.…”

Section: General Left-definite Theorymentioning

confidence: 99%

“…In this respect, we refer the reader to [3], [4], [5], [22], and [23] where general results on the Sobolev orthogonality of the Jacobi or Gegenbauer polynomials, when one or both parameters α and β are negative integers, are obtained. Bruder and Littlejohn [11] developed the spectral theory when α > −1 and β = −1. The analysis in [11] is similar in some respects to some of the results of this paper but, overall, quite different; whenever possible, we omit proofs which are similar to those given in [11].…”

Section: Introductionmentioning

confidence: 99%

“…Bruder and Littlejohn [11] developed the spectral theory when α > −1 and β = −1. The analysis in [11] is similar in some respects to some of the results of this paper but, overall, quite different; whenever possible, we omit proofs which are similar to those given in [11].…”

Section: Introductionmentioning

confidence: 99%

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Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

Bruder

Littlejohn

2012

Annali di Matematica

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In this paper, we consider the second-order differential expressionThis is the Jacobi differential expression with non-classical parameters α = β = −1 in contrast to the classical case when α, β > −1. For fixed k ≥ 0 and appropriate values of the spectral parameter λ, the equation ℓ[y] = λy has, as in the classical case, a sequence of (Jacobi) polynomial solutions {PThese Jacobi polynomial solutions of degree ≥ 2 form a complete orthogonal set in the Hilbert space L 2 ((−1, 1); (1 − x 2 ) −1 ). Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first degree solution, the set of polynomial solutions of degree ≥ 0 are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a selfadjoint operator T, generated by ℓ[·], in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman-Krein-Naimark theory is essential in helping to construct T in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.Date: May 22, 2012 (BL2v4.tex).

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Section: General Left-definite Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

Bruder

Littlejohn

2012

Annali di Matematica

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“…Special attention has been given to the case when u N is associated with the classical Jacobi or Laguerre orthogonal polynomials, and u k , 0 ≤ k ≤ N − 1, are derivatives of Dirac deltas supported on the endpoints of the interval of orthogonality of u N (see [1][2][3]5,15,18,19] and the references therein). Such bilinear forms arise when studying the orthogonality of Jacobi or Laguerre polynomials with the so-called non standard parameters, that is, negative integer parameters such that the coefficient c n in the corresponding three-term recurrence relation x p n (x) = a n p n+1 (x) + b n p n (x) + c n p n−1 (x), vanishes.…”

Section: Introductionmentioning

confidence: 99%

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

García-Ardila

Marriaga

2021

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

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Approximation by polynomials in Sobolev spaces associated with classical moment functionals

García-Ardila

Marriaga

2023

Numer Algor

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Let $$\textbf{u}$$ u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in $$H^r(\textbf{u})$$ H r ( u ) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the $$L^2$$ L 2 space associated with $$\textbf{u}$$ u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order $$N\leqslant r$$ N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of $$E_n(f^{(r)})$$ E n ( f ( r ) ) , the error of best approximation of $$f^{(r)}$$ f ( r ) in $$L^{2}(\textbf{u})$$ L 2 ( u ) .

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Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Cited by 4 publications

References 10 publications

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Approximation by polynomials in Sobolev spaces associated with classical moment functionals

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