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Orthogonal polynomials with asymptotically periodic recurrent coefficients. (English) Zbl 0604.42023

The authors take up the case of those class of orthogonal polynomials whose coefficients are asymptotically periodic (see, for example, T. J. Stieltjes [Ann. Fac. Sci. Toulouse 8J, 1-122 (1894), and 9A, 1-47 (1895)]). The measure associated with the three-term recurrence formula satisfies by these polynomials is constructed. They also discuss the asymptotic behavior of such polynomials.
Reviewer: A.N.Srivastava

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.)
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[1] Askey, R.; Ismail, M., Recurrence relations continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc., 300 (1984) · Zbl 0548.33001
[2] Case, K. M., Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys., 15, 2166-2174 (1974) · Zbl 0288.42009
[3] Case, K. M.; Kac, M., A discrete version of the inverse scattering problem, J. Math. Phys., 14, 594-603 (1973)
[4] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon & Breach: Gordon & Breach New York · Zbl 0389.33008
[5] Dombrowski, J., Spectral properties of phase operators, J. Math. Phys., 15, 576 (1974)
[6] Dombrowski, J.; Fricke, G. H., The absolute continuity of phase operators, Trans. Amer. Math. Soc., 213, 363-372 (1975) · Zbl 0346.47033
[8] Geronimo, J. S.; Case, K. M., Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Soc., 258, 467-494 (1980) · Zbl 0436.42018
[10] Mate, A.; Nevai, P.; Totik, V., Asymptotics for orthogonal polynomials defined by a recurrence relations, Constructive Approximation, 1, 231-248 (1985) · Zbl 0585.42023
[11] Nevai, P., Orthogonal polynomials, Mem. Amer. Math. Soc., 213 (1979) · Zbl 0405.33009
[12] Nevai, P., Orthogonal polynomials defined by a recurrence relation, Trans. Amer. Math. Soc., 250, 369-384 (1979) · Zbl 0413.42015
[13] Van Assche, W., Asymptotic properties of orthogonal polynomials from their recurrence formula I, J. Approx. Theory, 44, 258-276 (1985) · Zbl 0583.42011
[14] Stieltjes, T. J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 9, A1-A47 (1895)
[15] Perron, O., Die Lehre von den Kettenbrüchen (1950), Dekker: Dekker New York · JFM 55.0262.09
[16] Geronimus, Ya. L., On some finite difference equations and corresponding systems of orthogonal polynomials, Mem. Math. Sect. Fac. Math. Phys. Kharkov State Univ. and Kharkov Math. Soc., 25, 87-100 (1957), [Russian]
[17] Geronimus, Ya. L., On the character of the solutions of the moment problem in the case of a limit-periodic associated fraction, Bull. Acad. Sci. USSR, Sect. Math., 5, 203-210 (1941), [Russian] · Zbl 0060.26007
[18] Geronimus, Ya. L., Orthogonal polynomials, Amer. Math. Soc. Transl., 108, 37-130 (1977) · Zbl 0373.42007
[19] Szegö, G., Orthogonal polynomials, (Amer. Math. Soc. Colloq. Publ., Vol. 23 (1975), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · JFM 65.0278.03
[20] Barnsley, M. F.; Geronimo, J. S.; Harrington, A. N., Geometrical and electrical properties of some Julia sets, (Chudnovsky, D.; Chudnovsky, G., Classical and Quantum Models and Arithmetic Problems. Classical and Quantum Models and Arithmetic Problems, Lecture Notes in Pure and Applied Math., Vol. 92 (1984), Dekker: Dekker New York) · Zbl 0548.30021
[21] Freud, G., Orthogonal Polynomials (1971), Pergamon: Pergamon New York · Zbl 0226.33014
[22] Sario, L.; Nakai, M., Classification Theory of Riemann Surfaces (1970), Springer-Verlag: Springer-Verlag New York, Chap. III, Sect. 2 · Zbl 0199.40603
[23] Atkinson, F. V., Discrete and Continuous Boundary Value Problems (1964), Academic Press: Academic Press New York · Zbl 0117.05806
[24] Kac, M.; Van Moerbeke, P., On some periodic Toda lattices, (Proc. Nat. Acad. Sci. U.S.A., 72 (1975)), 1627-1629 · Zbl 0343.34003
[25] Van Moerbeke, P., The spectrum of periodic Jacobi matrices, Invent. Math., 37, 45-81 (1976) · Zbl 0361.15010
[26] Kato, K., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0148.12601
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