Orthogonal polynomials with respect to a symmetric inner product involving derivatives

Bavinck, H.; Meijer, H. G.

doi:10.1080/00036818908839864

Cited by 39 publications

(18 citation statements)

References 4 publications

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“…A survey of the results related to recurrence relations was given in [4]. Following the treatment [1], the nonsymmetric case concerning a unit point mass was studied in [5]. Orthogonal polynomials in a discrete space endowed with a nonstandard inner product were studied in [6].…”

Section: B;(1) -mentioning

confidence: 99%

“…(6) We are concerned with growth estimates of the polynomials B,,(z) on the interval [-1, 1]: In a certain sense we shall continue our study [1,2] of polynomials in Sobolev discrete spaces, where we obtained expressions for such polynomials and the distribution of their zeros. In a more general case these questions were studied in [3].…”

Section: B;(1) -mentioning

confidence: 99%

“…They were introduced in [1], where the Legendre polynomials e. = I --~g(n -2)(n -l)n(n + l)(n + 2)(n + 3).…”

Section: + N(f'(1)g'(1) +/'(-1)g'(-1))mentioning

confidence: 99%

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Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

Marcellán

Осиленкер

1997

Math Notes

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ABSTRACT. For the Legendre-Sobolev orthonormal polynomials B,t(z) = B,,(z ; M, N) depending on the parameters M > 0, N >_ 0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials B,t(z) for M > 0, N > 0 decay at the rate of n -z/2 at the points 1 and -1. The values of B'(:t:I) are calculated.KEY WORDS: Legendre-Sobolev polynomials, orthogonality with respect to the Legendre-Sobolev inner product, Legendre polynomials.

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Section: B;(1) -mentioning

confidence: 99%

Section: B;(1) -mentioning

confidence: 99%

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Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

Marcellán

Осиленкер

1997

Math Notes

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“…Meijer [3,4] in the (symmetric) Gegenbauer case α = β and for c = 1. In their second paper, they proved, in particular, that all zeros of π n (located, of course, symmetrically with respect to the origin) are real and simple, and that for n sufficiently large, there is exactly one pair of real zeros outside of the interval (-1,1).…”

Section: The Jacobi Weight Function Combined With Discrete Measuresmentioning

confidence: 99%

Computing orthogonal polynomials in Sobolev spaces

Gautschi

Zhang

1995

Numerische Mathematik

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Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other -a generalization of "Stieltjes's procedure" -expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Mathematics Subject Classification (1991): 65D15, 42C05

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“…Berti and Sri Ranga [3] and Berti, Bracciali and Sri Ranga [4] introduced an alternative approach to study Sobolev inner products with their orthogonal polynomials satisfying a recurrence relation of the form (1). This approach permits one to extend the pair of measures in the Sobolev inner products beyond coherent pairs and still maintain the required recurrence relation.…”

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confidence: 99%

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

2008

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Iserles et al. (J. Approx. Theory 65:151-175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115-140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.

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Orthogonal polynomials with respect to a symmetric inner product involving derivatives

Cited by 39 publications

References 4 publications

Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

Computing orthogonal polynomials in Sobolev spaces

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

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