Scattering theory and polynomials orthogonal on the real line
Authors:
J. S. Geronimo and K. M. Case
Journal:
Trans. Amer. Math. Soc. 258 (1980), 467494
MSC:
Primary 81F99; Secondary 30C10, 42C05
MathSciNet review:
558185
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Abstract: The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual threeterm recurrence formula, we derive a set of two twoterm recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as , to the solution of one of the recurrence formulas with the boundary conditions given at . In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.
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 Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory, translated from the Russian by B. D. Seckler, Gordon and Breach, New York, 1963. MR 0162497 (28:5696)
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 N. I. Akhiezer, The classical moment problem and some related questions in analysis, translated from the Russian by N. Kemmer, Hafner, New York, 1965. MR 0184042 (32:1518)
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 G. Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471487. MR 0126126 (23:A3422)
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 K. M. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 15 (1974), 21662174. MR 0353860 (50:6342)
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 K. M. Case and S. C. Chui, The discrete version of the Marchenko equation in the inverse scatering problem, J. Math. Phys. 14 (1973), 16431650. MR 0332067 (48:10394)
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 J. S. Geronimo, Scattering theory and orthogonal polynomials, Doctoral Dissertation, Rockefeller Univ., 1977.
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 J. S. Geronimo and K. M. Case, Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 201 (1979), 299. MR 519213 (80f:81100)
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 G. S. Guseinov, The determination of an infinite Jacobi matrix from the scattering data, Soviet Math. Dokl. 17 (1976), 596600. MR 0405160 (53:8955)
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 Paul Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc., no. 18, 1979, 185 pp. MR 519926 (80k:42025)
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 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939, p. 27; 4th edition, 1975.
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 A. Zygmund, Trigonometric series, vol. 1, 2nd ed., Cambridge Univ. Press, New York, 1959, pp. 245246. MR 0107776 (21:6498)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005581854
PII:
S 00029947(1980)05581854
Article copyright:
© Copyright 1980
American Mathematical Society
