Abstract
This paper studies the measure of orthogonality for a system of polynomials defined by a three term recursion formula, using the techniques of operator theory and functional analysis. Spectral properties of self-adjoint operators and compact operators, perturbation theorems, and commutator equations are used in the development of the ideas.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. I and II, Monographs and Studies in Mathematics, 9, 10, Pitman, London, 1981.
W. Bauldry, A. Mà té, and P. Nevai, Asymptotic Expansions of Recurrence Coefficients of Asymmetric Freud Polynomials, Approximation Theory V, New York: Academic Press, 251–254.
J. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Translations of Mathematical Monographs, 17, Amer. Math. Soc., Providence, 1968.
J. Dombrowski, Quasitriangular Matrices, Proc. Amer. Math. Soc. 69(1978), 95 – 96.
J. Dombrowski, Tridiagonal Matrix Representations of Cyclic Self-adjoint Operators, Pacific J. Math., 114(1984), 325 – 334.
J. Dombrowski, Tridiagonal Matrix Representations of Cyclic Self-adjoint Operators II, Pacific J. Math. 120 (1985), 47 – 53.
J. Dombrowski, Cyclic Operators, Commutators, and Absolutely Continuous Measures, Proc. Amer. Math. Soc. 100(1987), 457 – 463.
J. Dombrowski, Spectral Measures, Orthogonal Polynomials and Absolute Continuity, SIAM J. Math. Anal., 19(1988), 939 – 943.
J. Dombrowski, Spectral Measures Corresponding to Orthogonal Polynomials with Unbounded Recurrence Coefficients, Constr. Approx. 5(1989), 371 – 381.
J. Dombrowski and P. Nevai, Orthogonal Polynomials, Measures and Recurrence Relations, SIAM J. Math. Anal. 17(1986), 752 – 759.
T. Kato, Perturbation of Continuous Spectra by Trace Class Opera- torsProc. Japan Acad., 33(1957), 260 – 264.
A. Máté and P. Nevai, Orthogonal Polynomials and Absolutely Continuous Measures, Approximation Theory IV (C. Chui, L. L. Schumaker, J. D. Ward, eds.) New York: Academic Press (1983), 611 – 617.
A. Máté, P. Nevai and T. Zaslavsky, Asymptotic Expansion of Ratios of Coefficients of Orthogonal Polynomials With Exponential Weights, Trans. Amer. Math. Soc., 287 (1985), 495 – 505.
P. Nevia, Orthogonal Polynomials, Mem. Amer. Math. Soc., 213(1979).
C. R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Ergebnesse der Mathematik, 36, Springer, Berlin, 1967.
J. Shohat and J. Tamarkin, The Problem of Moments Mathematical Surveys, No. 1, Amer.Math. Soc., Providence, R. I., 1943/1950.
D. Sarason, Moment Problems and Operators in Hilbert Space, ‘Moments in Mathematics’Proceedings of Symposia in Applied Mathematics, 37, Amer. Math. Soc. 1987, 54 – 70.
M. Stone, Linear Transformations in Hilbert Space, Amer. Math. Soc., New York, 1932.
G Szegö, Orthogonal Polynomials, Amer. Math. Soc., Provindence, R.I., 1939, Fourth edition,1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Dombrowski, J. (1990). Orthogonal Polynomials and Functional Analysis. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_7
Download citation
DOI: https://doi.org/10.1007/978-94-009-0501-6_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6711-9
Online ISBN: 978-94-009-0501-6
eBook Packages: Springer Book Archive