Abstract
This paper surveys the theory of the three term recurrence relation for orthogonal polynomials and its relation to the spectral properties of the polynomials.
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References
Ahiezer, N. I. and Krein, M. G., ‘Some Questions in the Theory of Moments,’ Translations of Mathematical Monographs, vol. 2, Amer. Math. Soc., 1962.
Al-Salam, W. A. and Carlitz, L., ‘Some orthogonal q-polyomials,’ Math Nachr, 30(1965), 47 – 61.
Al-Salam, W. A. and Chihara, T. S., ‘Convolutions of orthogonal polynomials, ’ SIAM J Math. Anal., 7(1976), 16 – 28.
Askey, R. A. and Ismail, M. E. H., Recurrence relations, continued fractions and orthogonal polynomials, Memoirs of the Amer. Math. Soc., No. 300, 1984.
Askey, R. A. and Wilson, J. A., ‘A set of hypergeometric polynomials,’ SIAMJ. Math. Anal., 13 (1982), 651 – 655.
Askey, R. A. and Wilson,, Some basic hypergeometric orthogonal polynomials that generalize Jacobi Polynomials, Memoirs of the Amer. Math. Soc., No. 319, 1985.
Blumenthal, O., ‘Über die Entwicklung einer willkurlichen Funktion nach den Nennern des Kettenbruches für $$$$,’ Dissertation, Göttingen, 1898.
Chihara, T. S., ‘Convergent sequences of orthogonal polynomials,’ J. Math. Anal. Appl., 38(1972), 335–347.
Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, NewTork, 1978.
Chihara, T. S., ’Orthogonal polynomials whose distribution functions have finite point spectra,’ SIAM J. Math. Anal., 11(1980), 358–364.
Chihara, T. S., ’Orthogonal polynomials with discrete spectra on the real line,’ J. Approx. Theory, 42 (1984), 97–105.
Chihara, T. S., ’Hamburger moment problems and orthogonal polynomials,’ Trans. Amer. Math. Soc., to appear.
Chihara, T. S. and Ismail, M. E. H., ‘Orthogonal polynomials suggested by a queueing model, ’ Advances in Applied Math., 3(1982), 441 – 462.
Dennis, J. J. and Wall, H. S., ’The limit circle case for a positive-
definite J-fraction,’ Duke Math. J., 12(1945), 255–273.
Jacobsen, L. and Masson, D. R., ’On the convergence of limit periodic continued fractions K(an/1), where an ? -1/4. Part III.,Constructive Approximation, to appear.
Maki, D. P., ’A note on recursively defined orthogonal polynomials,’
Pacific J. Math., 28(1969), 611–613
Máté, A. and Nevai, P. G., ‘Orthogonal polynomials and absolutely continuous measures,’ in Approximation Theory, IV, C. K. Chui et al., eds., Academic Press, New York, 1983, 611 – 617.
Mátá, A., Nevai, P. G. and van Assche, W., ’The supports of measures associated with orthogonal polynomials and the spectra of the related self adjoint operators,’ Rocky Mtn. J. Math., to appear.
Nevai, P. G., Orthogonal Polyomials, Memoirs of Amer. Math. Soc., No. 213, 1979.
Nevai, P. G., ‘Two of my favorite ways of obtaining asymptotics for orthogonal polynomials, ’ International Series of Numerical Mathematics, 65(1984), 417 – 436.
Perron, O., Die Lehre von den Kettenbrüchen, 3rd ed., Leipzig, 1957.
Shohat, J. A. and Tamarkin, J. P., The Problem of Moments, Mathematical Surveys No. 1, Amer. Math. Soc., 1943.
Stieltjes, T. J., ’Recherches sur les fractions continues,’ Annales de la Facu1t’ des Sciences de Toulouse, 8 (1984), J 1–122; 9 (1895), A 1–47; Oeures,Vol. 2, 398–566.
Stone, M. H., Linear Transformations in Hilbert Spaces and Their Applications to Analysis,Colloquium Publications no. 15, Amer. Math. Soc., N.Y, 1932
Szegö, G. Orthogonal Polynomials, Colloquium Publications no. 23, Amer. Math. Soc., Providence, 4th edition, 1975.
Van Assche, W., ’The ratio of q-like orthogonal polynomials,’ J. Math. Anal. Appl., to appear.
Van Assche, W.,Asymptotics for Orthogonal Polynomials,Lecture Notes in Mathematics, 1265, Springer, Berlin, 1987.
Van Assche, W. and Geronimo, J.S., ‘Asymptotics for orthogonal polynomials on and off the essential spectrum,’ J. Approx Theory 55(1988), 220–231.
Van Doorn, E. A., ‘Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices,’J. Approx. Theory,51(1987), 254 – 266.
Wall, H. S., Continued Fractionsvan Nostrand, N. Y., 1948.
Wilson, J. A., ‘Some hypergeometric orthogonal polynomials, ’ SIAM J. Math. Anal., 11(1980), 690 – 701.
Wintner, A., Spektraltheorie der unendlichen Matrizen, 2nd ed., Leipzig, 1929.
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Chihara, T.S. (1990). The Three Term Recurrence Relation and Spectral Properties of Orthogonal Polynomials. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_4
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DOI: https://doi.org/10.1007/978-94-009-0501-6_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6711-9
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