×

On linearly related sequences of derivatives of orthogonal polynomials. (English) Zbl 1160.42011

An inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families \((P_n)_n\) and \((Q_n)_n\) whose derivatives of higher orders \(m\) and \(k\) (resp.) are connected by a linear algebraic structure relation such as \[ \sum^N_{i=0}r_{i,n}P_{n-i-m}^{(m)}(x)=\sum^M_{i=0} s_{i,n} Q^{(k)}_{n-i+k}(x) \] for all \(n=0,1,2,\dots\), where \(M\) and \(N\) are fixed nonnegative integer numbers, and \(r_{i,n}\) and \(s_{1,n}\) are given complex parameters satisfying some natural conditions. Let \(u\) and \(v\) be the moment regular functionals associated with \((P_n)_n\) and \((Q_n)_n\) (resp.). Assuming \(0\leq m\leq k\), we prove the existence of four polynomials \(\Phi_{M+m-i}\) and \(\Psi_{N+k+1}\), of degrees \(M+m+1\) and \(N+k+i\) (resp.), such that \[ D^{k-m}(\Phi_{M+m+i}u) =\Psi_{N+k+i}v(i=0,1), \] the \((k-m)\)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If \(k-m\), then \(u\) and \(v\) are connected by a rational modification. If \(k=m+1\), then both \(u\) and \(v\) are semiclassical linear functionals, which are also connected by a rational modification. When \(k+m\), the Stieltjes transform associated with \(u\) satisfies a non-homogeneous linear ordinary differential equation of order \(k-m\) with polynomial coefficients.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alfaro, M.; Marcellán, F.; Peña, A.; Rezola, M. L., On linearly related orthogonal polynomials and their functionals, J. Math. Anal. Appl., 287, 307-319 (2003) · Zbl 1029.42014
[2] Alfaro, M.; Marcellán, F.; Peña, A.; Rezola, M. L., On rational transformations of linear functionals: direct problem, J. Math. Anal. Appl., 298, 171-183 (2004) · Zbl 1066.33008
[3] Andrews, G. E.; Askey, R.; Roy, R., Special Functions, Encyclopedia Math. Appl., vol. 71 (1999), Cambridge University Press · Zbl 0920.33001
[4] Bonan, S.; Nevai, P., Orthogonal polynomials and their derivatives. I, J. Approx. Theory, 40, 134-147 (1984) · Zbl 0533.42015
[5] Bonan, S.; Lubinsky, D. S.; Nevai, P., Orthogonal polynomials and their derivatives. II, SIAM J. Math. Anal., 18, 4, 1163-1176 (1987) · Zbl 0638.42023
[6] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008
[7] Delgado, A. M.; Marcellán, F., Companion linear functionals and Sobolev inner products: a case study, Methods Appl. Anal., 11, 2, 237-266 (2004) · Zbl 1087.42020
[8] Iserles, A.; Koch, P. E.; Nørsett, S. P.; Sanz-Serna, J. M., On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory, 65, 2, 151-175 (1991) · Zbl 0734.42016
[9] Kwon, K. H.; Lee, J. H.; Marcellán, F., Generalized coherent pairs, J. Math. Anal. Appl., 253, 482-514 (2001) · Zbl 0967.33005
[10] Marcellán, F.; Branquinho, A.; Petronilho, J., On inverse problems for orthogonal polynomials. I, J. Comput. Appl. Math., 49, 153-160 (1993) · Zbl 0807.42017
[11] Marcellán, F.; Branquinho, A.; Petronilho, J., Classical orthogonal polynomials: a functional approach, Acta Appl. Math., 34, 3, 283-303 (1994) · Zbl 0793.33009
[12] Marcellán, F.; Martínez-Finkelshtein, A.; Moreno-Balcázar, J., \(k\)-Coherence of measures with non-classical weights, (Español, Luis; Varona, Juan L., Margarita Mathematica en Memoria de José Xavier Guadalupe Hernández (2001), Servicio de Publicaciones, Universidad de la Rioja: Servicio de Publicaciones, Universidad de la Rioja Logroño, Spain) · Zbl 1253.42022
[13] Marcellán, F.; Petronilho, J., On the solution of some distributional differential equations: existence and characterizations of the classical moment functionals, Integral Transforms Spec. Funct., 2, 3, 185-218 (1994) · Zbl 0832.33006
[14] Maroni, P., Sur quelques espaces de distributions qui sont des formes linéaires sur l’espace vectoriel des polynômes, (Brezinski, C.; etal., Simposium Laguerre, Bar-le-Duc. Simposium Laguerre, Bar-le-Duc, Lecture Notes in Math., vol. 1171 (1985), Springer-Verlag: Springer-Verlag Berlin), 184-194 · Zbl 0599.46051
[15] Maroni, P., Le calcul des formes linéaires et les polynômes orthogonaux semiclassiques, (Alfaro, M.; etal., Orthogonal Polynomials and Their Applications. Orthogonal Polynomials and Their Applications, Lecture Notes in Math., vol. 1329 (1988), Springer-Verlag: Springer-Verlag Berlin), 279-290 · Zbl 0661.42015
[16] Maroni, P., Une théorie algébrique des polynômes orthogonaux. Applications aux polynômes orthogonaux semiclassiques, (Brezinski, C.; etal., Orthogonal Polynomials and Their Applications. Orthogonal Polynomials and Their Applications, Proc. Erice, 1990. Orthogonal Polynomials and Their Applications. Orthogonal Polynomials and Their Applications, Proc. Erice, 1990, Ann. Comp. Appl. Math., vol. 9 (1991), IMACS), 95-130 · Zbl 0944.33500
[17] Maroni, P., Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math., 48, 133-155 (1993) · Zbl 0790.33006
[18] Medem, J. C., A family of singular semi-classical functionals, Indag. Math. (N.S.), 13, 3, 351-362 (2002) · Zbl 1031.42025
[19] Petronilho, J., Topological aspects in the theory of orthogonal polynomials and an inverse problem, (Bento, A.; etal., Proceedings of the Workshop on Analysis, The J.A. Sampaio Martins Anniversary Volume. Proceedings of the Workshop on Analysis, The J.A. Sampaio Martins Anniversary Volume, Textos Mat. Ser. B, vol. 34 (2004), Univ. Coimbra: Univ. Coimbra Coimbra), 91-107
[20] Petronilho, J., On the linear functionals associated to linearly related sequences of orthogonal polynomials, J. Math. Anal. Appl., 315, 379-393 (2006) · Zbl 1084.42020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.