On an extension of symmetric coherent pairs of orthogonal polynomials

Delgado, Antonia M.; Marcellán, Francisco

doi:10.1016/j.cam.2004.02.026

Cited by 15 publications

(15 citation statements)

References 5 publications

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“…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. de Jesus and J. Petronilho [14,15], A. Branquinho and 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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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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“…(9) is an important tool to study such Sobolev orthogonal polynomials. Indeed, in [28], it was proven that if (9) holds, then:…”

Section: Introductionmentioning

confidence: 99%

“…with c n = 0, for n ≥ 2. Some examples of symmetric measures whose sequences of orthogonal polynomials satisfy (10) have been studied in [28]. Asymptotic properties of the corresponding sequences of orthogonal polynomials and the location of their zeros were analyzed in [29,30] for the Gegenbauer case, as well as in [31,32] for the Hermite case.…”

Section: Introductionmentioning

confidence: 99%

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

2019

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“…About this subject, in [4] is presented the algebraic relation between the Sobolev polynomials and the polynomials {R n } n∈N ,; besides, the particular case where V is classical is studied, and the respective symmetric (1, 1)−coherent companion is found. In particular, we focus on the symmetric (1, 1)−coherent pair…”

Section: Introductionmentioning

confidence: 99%

On some asymptotic properties of classical Hermite polynomials modified by a rational factor

Molano

2018

revint

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In this paper we study some asymptotic properties of the sequence of monic polynomials orthogonal with respect to the measure dµ = x 2 +a x 2 +b e −x 2 dx, where a, b > 0 and a = b. In this way we study the outer relative asymptotic with respect to the classical Hermite polynomials; besides, Mehler-Heine type formulas are analyzed.Sobre algunas propiedades asintóticas de polinomios de Hermite clásicos modificados por un factor racional Resumen. En este artículo estudiamos algunas propiedades asintóticas de la sucesión de polinomios mónicos ortogonales con respecto a la medida dµ = x 2 +a x 2 +b e −x 2 dx, donde a, b > 0 y a = b. En este sentido, estudiamos la asintótica relativa exterior con respecto a los polinomios clásicos de Hermite, además son analizadas fórmulas tipo Mehler-Heine.

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On an extension of symmetric coherent pairs of orthogonal polynomials

Cited by 15 publications

References 5 publications

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

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

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

On some asymptotic properties of classical Hermite polynomials modified by a rational factor

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