Zeros of Sobolev orthogonal polynomials following from coherent pairs

Meijer, H. G.; Bruin, Marcel G. de

doi:10.1016/s0377-0427(01)00421-6

Cited by 16 publications

(21 citation statements)

References 5 publications

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“…and so, the change α → α − 1 must be made in the results of [36] for case I to agree with the notation (3). Let Q (L1) n be the Laguerre-Sobolev orthogonal polynomials of case I, i.e., orthogonal with respect to Sobolev inner products induced by the measures given by (3) or (4), then for n ≥ 2,…”

Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

“…In [36] a complete study of the zeros of coherent pairs of Jacobi and Laguerre types using Gaussian quadrature formulas was done. Before stating the results it is important to point out that the notation used in [36] for Laguerre-Sobolev of case I is:…”

Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

“…Before stating the results it is important to point out that the notation used in [36] for Laguerre-Sobolev of case I is:…”

Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

“…Finally, (40) follows from (36) and applying Lemma 14.6 in [21, p.395] with p = 2, m = n − 1, and w = W n . To establish (41) we need this Lemma 14.6, (45), (46), and some other technical results.…”

Section: (C)]mentioning

confidence: 99%

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Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

Marcellán

Moreno-Balcázar

2006

Acta Appl Math

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In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.MSC: Primary 42C05; Secondary 33C45

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Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

“…Before stating the results it is important to point out that the notation used in [36] for Laguerre-Sobolev of case I is:…”

Section: Zeros and Its Asymptoticsmentioning

confidence: 99%

Section: (C)]mentioning

confidence: 99%

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Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

Marcellán

Moreno-Balcázar

2006

Acta Appl Math

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“…Lee, and F. Marcellán [4], F. Marcellán, A. Martínez-Finkelshtein and J.J. Moreno-Balcázar [5], M. de Bruin and H.G. Meijer [6], M. Alfaro, F. Marcellán, A. Peña, and M.L. Rezola [7][8][9][10], A.M. Delgado and F. Marcellán [11,12], J. Petronilho [13], M.N.…”

Section: Introductionmentioning

confidence: 99%

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

Jesus¹,

Marcellán²,

Petronilho³

et al. 2014

Journal of Computational and Applied Mathematics

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a b s t r a c tA pair of regular linear functionals (U, V) is said to be a (M, N)-coherent pair of order (m, k) if their corresponding sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 satisfy a structure relation such as To be more precise, let us assume that m ≥ k. If m = k then U and V are related by a rational factor (in the distributional sense); if m > k then U and V are semiclassical and they are again related by a rational factor. In the second part of this work we deal with a Sobolev type inner product defined in the linear space of polynomials with real coefficients, P, aswhere λ is a positive real number, m is a positive integer number and (μ 0 , μ 1 ) is a (M, N)-coherent pair of order m of positive Borel measures supported on an infinite subset of the real line, meaning that the sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 with respect to μ 0 and μ 1 , respectively, satisfy a structure relation as above with k = 0, a i,n and b i,n being real numbers fulfilling the above mentioned conditions. We generalize several recent results known in the literature in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs (introduced in [A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products J. Approx. Theory 65 (2) (1991) 151-175]) and their extensions. In particular, we show how to compute the coefficients of the Fourier expansion of functions on an appropriate Sobolev space (defined by the above inner product) in terms of the sequence of Sobolev orthogonal polynomials {S n (x; λ)} n≥0 .

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On generating Sobolev orthogonal polynomials

Van Buggenhout

2023

Numer. Math.

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Zeros of Sobolev orthogonal polynomials following from coherent pairs

Cited by 16 publications

References 5 publications

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

On generating Sobolev orthogonal polynomials

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