SOBRE LAS RELACIONES DE RECURRENCIA, LAS FRACCIONES CONTINUAS Y LA DETERMINACION DE LAS PROPIEDADES ESPECTRALES DE LOS SISTEMAS ORTOGONALES DE POLINOMIOS.

Jairo A.  Charris; Bernarda H.  Aldana; Germán  Preciado

doi:10.18257/raccefyn.27(104).2003.2077

Vol. 27 No. 104 (2003), Mathematics

Vol. 27 No. 104 (2003)

ON THE RECUNRRENCE RELATIONS, THE CONTINUED FRACTIONS AND THE DETERMINATION OF THE SPECTRAL PROPERTIES OF ORTHOGONAL POLYNOMIALS

Mathematics

https://doi.org/10.18257/raccefyn.27(104).2003.2077

Published 2023-10-11

Jairo A. Charris⁺⁻
Bernarda H. Aldana⁺⁻
Germán Preciado⁺⁻

Jairo A. Charris

Universidad Sergio Arboleda y Academia Colombiana de Ciencias Exactas, Físicas y Naturales, Bogotá D.C.

Bernarda H. Aldana

Escuela Colombiana de Ingeniería.

Germán Preciado

Universidad Nacional de Colombia y Universidad de los Andes, Bogotá, D.C.

PDF (Español (España))

How to Cite

Charris, J. A. ., Aldana, B. H. ., & Preciado, G. . (2023). ON THE RECUNRRENCE RELATIONS, THE CONTINUED FRACTIONS AND THE DETERMINATION OF THE SPECTRAL PROPERTIES OF ORTHOGONAL POLYNOMIALS. Revista De La Academia Colombiana De Ciencias Exactas, Físicas Y Naturales, 27(104), 381–421. https://doi.org/10.18257/raccefyn.27(104).2003.2077

Downloads

Download data is not yet available.

Métricas Alternativas

Dimensions

Abstract

This half-research, half-expository paper addresses the problem of determining the spectrum and spectral properties of systems of orthogonal polynomials that can be derived from their recurrence relation or, more precisely, from the continued fraction of the polynomials. The content has been the subject of recent research of the authors (and of many others as well), but this time the paper explores widely the foundational basis of the subject.

https://doi.org/10.18257/raccefyn.27(104).2003.2077

Keywords

Orthogonal polynomials | spectrum | spectral properties of systems of orthogonal polynomials | recurrence relations | continued fractions

PDF (Español (España))

References

I. N. Ahkiezer, The Classical Moment Problem. Hafner, New York, 1965.

B. H. Aldana, J.A. Charris & O. Mora–Valbuena, On block recursions, Askey’s sieved Jacobi polynomials, & two related systems. Colloquium Mathematicum 78 (1998), 57–91.

T. M. Apostol, Mathematical Analysis. 2nd Edition, Addison Wesley, Reading, Mass., 1974.

W. Al–Salam, W. Allaway & R. Askey, Sieved ultraspherical polynomials. Trans. Amer. Math. Soc. 234 (1984), 39–55.

R. Askey, Orthogonal polynomials old & new & some combinatorial connections. In: Enumeration & Design, D. M. Jackson & S. A. Vanstone (eds.), Academic Press, Toronto, 1984, 67–84.

R. Askey, & M. E. H. Ismail, Recurrence relations, continued fractions & orthogonal polynomials. Mem. Amer. Math. Soc. 300 (1984), 1–102.

E. Banck & M. E. H. Ismail, The attractive Coulomb potential polynomials. Constructive Approximation 1 (1985), 103.

R. Bartle, The Elements of Integration & Lebesgue Measure. Wiley, New York, 1995.

TJ. T. Broad, Gauss quadrature generated by diagonalization of H in infinite L2 basis. Phys Rev. A–18, 1012–1027.

J. A. Charris & L. A. G ́omez, Functional analysis, orthogonal polynomials & a theorem of Markov. Revista Colombiana Mat. 22 (1988), 79–128.

J. A. Charris & M. E. H. Ismail, Sieved orthogonal polynomials VII: Generalized polynomial mappings. Trans. Amer. Math. Soc., 340 (1993), 71–93.

J. A. Charris, M. E. H. Ismail, & S. Monsalve, Sieved orthogonal polynomials X: General blocks of recurrence relations. Pac. J. Math. 163 (1994), 1294–1308.

J. A. Charris & G. Rodr ́ıguez–Blanco, On systems of orthogonal polynomials with inner & end point masses. Revista Colombiana Mat. 24 (1990), 153–177.

J. A. Charris & F. H. Soriano, Complex & distributional weights for sieved ultraspherical polynomials. Internat. J. Math. & Math. Sci. 19 (1996), 229 - 242.

J. A. Charris & F. H. Soriano, On the distributional orthogonality of the general Pollaczek polynomials. Internat. J. Math. & Math. Sci. 19 (1996), 417–426. Gordon and Breach, New York, 1978.

J. A. Charris & O. Mora–Valbuena, On block recursions & the determination of spectral measures from continued fractions. Int. J. Appl. Math. 1 (1999), 635–688.

J. A. Charris & G. Preciado–L ́opez, Sobre los polinomios ortogonales, las fracciones continuas y las medidas espectrales. Rev. Acad. Col. Cienc. Exactas F ́ısicas y Naturales 26 (2002), 403–410.

T. S. Chihara, On co-recursive orthogonal polynomials. Proc. Amer. Math. Soc. 8(1957), 899–905.

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

J. Ger ́onimo & W. van Assche, Orthogonal polynomials on several intervals via a polynomial mapping. Trans Amer. Math. Soc. 308 (1988), 559–581.

E. J. Heller, W. P. Reinhard & H. A. Yamani, On quadrature calculations of matrix elements using L2–expansion techniques. J. Comp. Phys. 13 (1973), 535–549.

M. E. H. Ismail, D. Masson & M. Rahman, Complex weight functions for classical orthogonal polynomials. Canadian J. Mathematics 43 (1991), 1294–1308.

A. Krall, Orthogonal polynomials through moment functionals. SIAM J. Math. Anal. 9 (1978), 600–603.

F. W. Olver, Asymptotics & Special Functions. Academic Press, New York, 1974.

E. D. Rainville, Special Functions. Macmillan, New York, 1960.

W. Rudin, Real & Complex Analysis. 2nd Edition, Mc Graw-Hill, New York, 1974.

J. Shohat & J. P. Tamarkin, The problem of Moments. Math. Surveys Vol. 1, Amer. Math. Soc, Providence, R.I., 1950.

G. B. Simmons, Introduction to Topology & Modern Analysis. Mc Graw-Hill, New York, 1963.

G. Szeg ̈o, Orthogonal Polynomials. 4th Ed., Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1975.

H. S. Wall, Analytic Theory of Continued Fractions. Van Nostrand, New York, 1948.

H. A. Yamani & W. P. Reinhardt, L2 discretization of the continuum, radial kinetic energy & Coulomb hamiltonians. Phys. Rev., A–11, 1144–1155.

K. Yosida, Functional Analysis. 5th Edition, Springer, Berlín, 1979.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.