On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

Herbert Dueñas Ruiz; Francisco Marcellán; Alejandro Molano

doi:10.15446/recolma.v53n2.85524

Published

2019-07-01

On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

Sobre (1, 1) pares coherentes simétricos y polinomios ortogonales Sobolev: un algoritmo para calcular coeficientes de Fourier

DOI:

https://doi.org/10.15446/recolma.v53n2.85524

Keywords:

Orthogonal polynomials, Symmetric (1 1)-coherent pairs, Sobolev-Fourier series (en)
Polinomios ortogonales, Pares Simétricos (1 1)-Coherentes, Series Sobolev-Fourier (es)

Downloads

PDF (Español)

Authors

Herbert Dueñas Ruiz Universidad Nacional de Colombia
Francisco Marcellán Universidad Carlos III de Madrid and Instituto de Ciencias Matemáticas (ICMAT)
Alejandro Molano Universidad Pedagógica y Tecnológica de Colombia

Abstract (en)
Abstract (es)

In the pioneering paper [13], the concept of Coherent Pair was introduced by Iserles et al. In particular, an algorithm to compute Fourier Coefficients in expansions of Sobolev orthogonal polynomials defined from coherent pairs of measures supported on an infinite subset of the real line is described. In this paper we extend such an algorithm in the framework of the so called Symmetric (1, 1)-Coherent Pairs presented in [8].

En el artículo pionero [13], fue introducido el concepto de Par Coherente por Iserles et al. En particular, allí es descrito un algoritmo para calcular coeficientes de Fourier de expansiones de polinomios ortogonales de tipo Sobolev definidos a partir de pares de medidas coherentes soportadas en un subconjunto infinito de la recta real. En esta contribución extendemos tal algoritmo en el contexto de los llamados Pares Simétricos (1, 1)-Coherentes presentados en [8].

References

P. Althammer, Eine erweiterung des orthogonalitatsbegriffes bei polynomen und deren anwendung auf die beste approximation, J. Reine Angew. Math. 211 (1962), 192-204.

A. C. Berti and A. Sri Ranga, Companion orthogonal polynomials: some applications, Appl. Numer. Math. 39 (2001), 127-149.

C. Brezinski, A direct proof of the Christoffel-Darboux identity and its equivalence to the recurrence relationship, J Comput Appl Math. 32 (1990), no. 1, 17-25.

T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.

M. N. de Jesús, F. Marcellán, J. Petronilho, and N. Pinzón-Cortés, (M, N)-coherent pairs of order (m, k) and Sobolev orthogonal polynomials, J. Comput. Appl. Math. 256 (2014), 16-35.

M. N. de Jesús and J. Petronilho, On linearly related sequences of derivatives of orthogonal polynomials, J. Math. Anal. Appl. 347 (2008), 482-492.

A. M. Delgado, Ortogonalidad no estándar: problemas directos e inversos, Tesis doctoral, Universidad Carlos III de Madrid, 2006.

A. M. Delgado and F. Marcellán, On an extension of symmetric coherent pairs of orthogonal polynomials, J. Comput. Appl. Math. 178 (2005), 155-168.

H. Dueñas, F. Marcellán, and A. Molano, Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs, J. Math. Anal. Appl. 467 (2018), no. 1, 601-621.

H. Dueñas, F. Marcellán, and A. Molano, A classification of symmetric (1, 1)-coherent pairs, Mathematics 7 (2019), no. 2-213, 1-32.

J. Favard, Sur les polynômes de Tchebicheff, C.R. Acad. Sci. Paris 200 (1935), 2052-2053.

B. Fischer and G. H. Golub, How to generate unknown orthogonal polynomials out of known orthogonal polynomials, J. Comput. Appl. Math. 43 (1992), 99-115.

A. Iserles, P. E. Koch, S. P. Norsett, and J. M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), 151-175.

M. Ismail and D. R. Masson, Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), 1-40.

D. C. Lewis, Polynomial least square approximations, Amer. J. Math. 69 (1947), 273-278.

F. Marcellán and R. Álvarez Nodarse, On the "Favard theorem" and its

extensions, J. Comput. Appl. Math. 127 (2001), 231-254.

F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math. 33 (2015), no. 3, 308-352.

P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, In Orthogonal polynomials and their applications (Erice, 1990), C. Brezinski, L. Gori, A, Ronveaux Editors, IMACS Ann. Comput. Appl. Math. 9 (1991), 95-130.

H. G. Meijer, A short history of orthogonal polynomials in a Sobolev space I. The non-discrete case, Niew Archief voor Wiskunde 14 (1996), 93-113.

How to Cite

APA

Dueñas Ruiz, H., Marcellán, F. and Molano, A. (2019). On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients. Revista Colombiana de Matemáticas, 53(2), 139–164. https://doi.org/10.15446/recolma.v53n2.85524

ACM

[1]

Dueñas Ruiz, H., Marcellán, F. and Molano, A. 2019. On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients. Revista Colombiana de Matemáticas. 53, 2 (Jul. 2019), 139–164. DOI:https://doi.org/10.15446/recolma.v53n2.85524.

ACS

(1)

Dueñas Ruiz, H.; Marcellán, F.; Molano, A. On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients. rev.colomb.mat 2019, 53, 139-164.

ABNT

DUEÑAS RUIZ, H.; MARCELLÁN, F.; MOLANO, A. On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients. Revista Colombiana de Matemáticas, [S. l.], v. 53, n. 2, p. 139–164, 2019. DOI: 10.15446/recolma.v53n2.85524. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/85524. Acesso em: 22 jan. 2025.

Chicago

Dueñas Ruiz, Herbert, Francisco Marcellán, and Alejandro Molano. 2019. “On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients”. Revista Colombiana De Matemáticas 53 (2):139-64. https://doi.org/10.15446/recolma.v53n2.85524.

Harvard

Dueñas Ruiz, H., Marcellán, F. and Molano, A. (2019) “On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients”, Revista Colombiana de Matemáticas, 53(2), pp. 139–164. doi: 10.15446/recolma.v53n2.85524.

IEEE

[1]

H. Dueñas Ruiz, F. Marcellán, and A. Molano, “On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients”, rev.colomb.mat, vol. 53, no. 2, pp. 139–164, Jul. 2019.

MLA

Dueñas Ruiz, H., F. Marcellán, and A. Molano. “On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients”. Revista Colombiana de Matemáticas, vol. 53, no. 2, July 2019, pp. 139-64, doi:10.15446/recolma.v53n2.85524.

Turabian

Dueñas Ruiz, Herbert, Francisco Marcellán, and Alejandro Molano. “On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients”. Revista Colombiana de Matemáticas 53, no. 2 (July 1, 2019): 139–164. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/85524.

Vancouver

1.

Dueñas Ruiz H, Marcellán F, Molano A. On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients. rev.colomb.mat [Internet]. 2019 Jul. 1 [cited 2025 Jan. 22];53(2):139-64. Available from: https://revistas.unal.edu.co/index.php/recolma/article/view/85524

Download Citation

CrossRef Cited-by

4

1. Luis Alejandro Molano Molano. (2023). Inverse problems, Sobolev–Chebyshev polynomials and asymptotics. Ukrains’kyi Matematychnyi Zhurnal, 75(10), p.1411. https://doi.org/10.3842/umzh.v75i10.7293.

2. G. A. Marcato, F. Marcellán, A. Sri Ranga, Yen Chi Lun. (2023). Coherent pairs of measures of the second kind on the real line and Sobolev orthogonal polynomials. An application to a Jacobi case. Studies in Applied Mathematics, 151(2), p.475. https://doi.org/10.1111/sapm.12583.

3. M. Hancco Suni, G.A. Marcato, F. Marcellán, A. Sri Ranga. (2023). Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials. Journal of Mathematical Analysis and Applications, 525(1), p.127118. https://doi.org/10.1016/j.jmaa.2023.127118.

4. Luis Alejandro Molano Molano. (2024). Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics. Ukrainian Mathematical Journal, 75(10), p.1601. https://doi.org/10.1007/s11253-024-02281-3.