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 three-term recurrence formula, we derive a set of two two-term 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 $n \to \infty$, to the solution of one of the recurrence formulas with the boundary conditions given at $n = 0$. 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.
- Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory, Gordon and Breach Science Publishers, New York-London, 1963. Translated from the Russian by B. D. Seckler. MR 0162497
- N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
- Glen Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471–487. MR 126126, DOI 10.1090/S0002-9947-1961-0126126-8
- K. M. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Mathematical Phys. 15 (1974), 2166–2174. MR 353860, DOI 10.1063/1.1666597
- K. M. Case and S. C. Chiu, The discrete version of the Marchenko equations in the inverse scattering problem, J. Mathematical Phys. 14 (1973), 1643–1647. MR 332067, DOI 10.1063/1.1666237 J. S. Geronimo, Scattering theory and orthogonal polynomials, Doctoral Dissertation, Rockefeller Univ., 1977.
- J. S. Geronimo and K. M. Case, Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 20 (1979), no. 2, 299–310. MR 519213, DOI 10.1063/1.524077
- G. Š. Guseĭnov, Determination of an infinite Jacobi matrix from scattering data, Dokl. Akad. Nauk SSSR 227 (1976), no. 6, 1289–1292 (Russian). MR 0405160
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939, p. 27; 4th edition, 1975.
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 258 (1980), 467-494
- MSC: Primary 81F99; Secondary 30C10, 42C05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0558185-4
- MathSciNet review: 558185