Sobolev Orthogonal Polynomials and Second-Order Differential Equations

Kwon, Kil Hyun; Littlejohn, Lance L.

doi:10.1216/rmjm/1181071786

Cited by 50 publications

(36 citation statements)

References 15 publications

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“…we obtain the following theorem; a proof can be found in [12]. A key result in establishing this result is in the following important identity…”

Section: The Sobolev Orthogonality Of the Jacobi Polynomialsmentioning

confidence: 79%

“…Moreover, by the GKN theory, there exists a self-adjoint differential operator A (α,−1) , generated by l α,−1 [·], that is positively bounded below in L 2 α,−1 (−1, 1), that has the set {P (α,−1) n } ∞ n=1 as eigenfunctions. Furthermore, in [12], Kwon In this paper we prove that the entire sequence of Jacobi polynomials {P (α,−1) n } ∞ n=0 are, in fact, complete in W α . More importantly, we construct a self-adjoint, positively bounded below operator T α , generated from l α,−1 [·], in W α having the entire set of Jacobi polynomials {P (α,−1) n } ∞ n=0 as a complete set of eigenfunctions.…”

Section: Introductionmentioning

confidence: 89%

“…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%

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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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“…we obtain the following theorem; a proof can be found in [12]. A key result in establishing this result is in the following important identity…”

Section: The Sobolev Orthogonality Of the Jacobi Polynomialsmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 89%

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Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Bruder

Littlejohn

2011

Results. Math.

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“…We refer the reader to [16,21], for the case of non-classical Laguerre families; [17,3,2,1] for non-standard orthogonality concerning Jacobi polynomials; [4,10] for the case of Meixner polynomials with non-standard parameters; [7,8], for the case of symmetric Meixner-Pollaczek polynomials with parameters out of classical considerations; [11], for (not necessarily symmetric) generalized Meixner-Pollaczek polynomials with null parameter λ. .…”

Section: Non-standard Orthogonalitymentioning

confidence: 99%

Non-classical orthogonality relations for big and little q-Jacobi polynomials

Moreno

García−Caballero

2010

Journal of Approximation Theory

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Big q-Jacobi polynomials {P n (·; a, b, c; q)} ∞ n=0 are classically defined for 0 < a < q −1 , 0 < b < q −1 and c < 0. For the family of little q-Jacobi polynomials { p n (·; a, b|q)} ∞ n=0 , classical considerations restrict the parameters imposing 0 < a < q −1 and b < q −1 . In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditions for those cases for which Favard's theorem does not work. As a by-product, we obtain similar results for the families of big and little q-Laguerre polynomials.

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“…The orthogonality of some classical systems of polynomials, in the outstanding situation in which the three term recurrence relation breaks down (so the hypothesis of Favard's theorem does not hold), has been successfully developed in the last decade: see [21,24] for the Laguerre case, [22,4,2,1] for the Jacobi case, [3,12] for the Meixner case, [5,6] for the case of symmetric Meixner-Pollaczek polynomials, [13] for (not necessarily symmetric) Meixner-Pollaczek polynomials with null parameter and, finally, [12] for the classical families of polynomials which satisfy a discrete orthogonality with a finite number of masses (i.e., the Hahn, Racah, dual Hahn and Krawtchouk polynomials).…”

Section: Introductionmentioning

confidence: 99%

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

Moreno

García−Caballero

2010

Journal of Mathematical Analysis and Applications

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Keywords:Continuous q-ultraspherical polynomials Discrete q-ultraspherical polynomials Non-standard orthogonality Favard's theorem In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big qJacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.

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Sobolev Orthogonal Polynomials and Second-Order Differential Equations

Cited by 50 publications

References 15 publications

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Non-classical orthogonality relations for big and little q-Jacobi polynomials

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

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