×

\(\Delta\)-coherent pairs and orthogonal polynomials of a discrete variable. (English) Zbl 1047.42019

The concept of coherent pairs of measures (or linear functionals) turned out to be a useful tool in the study of Sobolev orthogonal polynomials (that is, orthogonal with respect to inner product involving derivatives). A natural generalization of this concept is obtained by replacing derivative by its finite-difference approximation. The authors extend the notion of coherence to this inner product, and prove that under certain conditions every coherent pair must contain a classical discrete linear functional, like in the case of continuous coherent pairs.

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

References:

[1] Abdelkarim F., Result. Math. 32 pp 1– (1997)
[2] Abramowitz M., Handbook of Mathematical Functions (1965)
[3] DOI: 10.1080/10652469508819081 · Zbl 0849.33007 · doi:10.1080/10652469508819081
[4] DOI: 10.1016/0377-0427(95)00097-6 · Zbl 0865.42023 · doi:10.1016/0377-0427(95)00097-6
[5] DOI: 10.1080/10652460008819238 · Zbl 0972.42017 · doi:10.1080/10652460008819238
[6] Chihara T. S., An Introduction to Orthogonal Polynomials (1978) · Zbl 0389.33008
[7] DOI: 10.1016/0377-0427(93)E0241-D · Zbl 0853.33009 · doi:10.1016/0377-0427(93)E0241-D
[8] Gautschi W., International Series of Numerical Mathematics 57, in: Numerical Integration. Proceedings of the Conference 1981 Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4-10 pp 89– (1982)
[9] DOI: 10.1016/0021-9045(91)90100-O · Zbl 0734.42016 · doi:10.1016/0021-9045(91)90100-O
[10] Koekoek R., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Fac. Techn. Math (1998)
[11] Lesley P. A., Sitzungsber. Abt. 11 pp 204– (1995)
[12] DOI: 10.1109/18.412678 · Zbl 0836.94025 · doi:10.1109/18.412678
[13] DOI: 10.1007/BF01759996 · Zbl 0771.33008 · doi:10.1007/BF01759996
[14] DOI: 10.1016/S0377-0427(97)00057-5 · Zbl 0885.42013 · doi:10.1016/S0377-0427(97)00057-5
[15] DOI: 10.1016/0019-3577(95)93197-I · Zbl 0843.42010 · doi:10.1016/0019-3577(95)93197-I
[16] DOI: 10.1016/0377-0427(95)00121-2 · Zbl 0855.42016 · doi:10.1016/0377-0427(95)00121-2
[17] DOI: 10.1080/10236199808808156 · Zbl 0916.33006 · doi:10.1080/10236199808808156
[18] Maroni P., Orthogonal Polynomials and TheirApplications 9 pp 95– (1991)
[19] DOI: 10.1006/jath.1997.3123 · Zbl 0898.42006 · doi:10.1006/jath.1997.3123
[20] DOI: 10.1006/jath.1996.3062 · Zbl 0880.42012 · doi:10.1006/jath.1996.3062
[21] Nikiforov A. F., Classical Orthogonal Polynomials of a DiscreteVariable (1991) · doi:10.1007/978-3-642-74748-9
[22] DOI: 10.1090/S0002-9939-98-04300-7 · Zbl 0895.42009 · doi:10.1090/S0002-9939-98-04300-7
[23] DOI: 10.1016/S0377-0427(96)00168-9 · Zbl 0871.42024 · doi:10.1016/S0377-0427(96)00168-9
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.