Orthogonal polynomials, measures and recurrences on the unit circle
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- by Paul Nevai
- Trans. Amer. Math. Soc. 300 (1987), 175-189
- DOI: https://doi.org/10.1090/S0002-9947-1987-0871671-9
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Abstract:
New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials.References
- N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965.
- Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
- 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
- A. Bultheel, Convergence of Schur parameters and transmission zeros of a meromorphic spectrum, Computational and combinatorial methods in systems theory (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 281–296. MR 924013
- K. M. Case, Orthogonal polynomials revisited, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 289–304. MR 0390322
- 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
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- Philippe Delsarte and Yves V. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process. 34 (1986), no. 3, 470–478. MR 844658, DOI 10.1109/TASSP.1986.1164830 —, Application of the split Levinson algorithm: the ultraspherical, polynomials, manuscript.
- J. Dombrowski, Spectral properties of phase operators, J. Mathematical Phys. 15 (1974), 576–577. MR 334757, DOI 10.1063/1.1666686
- J. Dombrowski, Spectral properties of real parts of weighted shift operators, Indiana Univ. Math. J. 29 (1980), no. 2, 249–259. MR 563209, DOI 10.1512/iumj.1980.29.29016
- Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114 (1984), no. 2, 325–334. MR 757504
- J. Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators. II, Pacific J. Math. 120 (1985), no. 1, 47–53. MR 808928
- J. Dombrowski and G. H. Fricke, The absolute continuity of phase operators, Trans. Amer. Math. Soc. 213 (1975), 363–372. MR 377573, DOI 10.1090/S0002-9947-1975-0377573-0
- Joanne Dombrowski and Paul Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17 (1986), no. 3, 752–759. MR 838253, DOI 10.1137/0517054 G. Freud, Orthogonal polynomials, Pergamon Press, New York, 1971.
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
- Jeffrey S. Geronimo, Matrix orthogonal polynomials on the unit circle, J. Math. Phys. 22 (1981), no. 7, 1359–1365. MR 626123, DOI 10.1063/1.525073
- 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
- J. S. Geronimo and K. M. Case, Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Soc. 258 (1980), no. 2, 467–494. MR 558185, DOI 10.1090/S0002-9947-1980-0558185-4
- L. Ya. Geronimus, Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Consultants Bureau, New York, 1961. Authorized translation from the Russian. MR 0133643 —, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. 3 (1962), 1-78. —, On asymptotic properties of polynomials which are orthogonal on the unit circle, and on certain properties of positive harmonic functions, Amer. Math. Soc. Transl. 3 (1962), 79-106. —, Orthogonal polynomials, Amer. Math. Soc. Transl. 108 (1977), 37-130.
- G. L. Lopes, Asymptotic behavior of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Mat. Sb. (N.S.) 128(170) (1985), no. 2, 216–229, 287 (Russian). MR 809486 Al. Magnus, Reflection coefficients of the circular Jacobi polynomials, private communication.
- Attila Máté and Paul G. Nevai, Bernstein’s inequality in $L^{p}$ for $0<p<1$ and $(C,\,1)$ bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), no. 1, 145–154. MR 558399, DOI 10.2307/1971219
- Attila Máté and Paul G. Nevai, Remarks on E. A. Rakhmanov’s paper: “The asymptotic behavior of the ratio of orthogonal polynomials” [Mat. Sb. (N.S.) 103(145) (1977), no. 2, 237–252; MR 56 #3556], J. Approx. Theory 36 (1982), no. 1, 64–72. MR 673857, DOI 10.1016/0021-9045(82)90071-5
- Attila Máté and Paul Nevai, Orthogonal polynomials and absolutely continuous measures, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 611–617. MR 754400 A. Máté, P. Nevai, and V. Totik, What is beyond Szegö’s theory of orthogonal polynomials, Rational Approximation and Interpolation (P. R. Graves-Morris et al., eds.), Lecture Notes in Math., vol. 1105, Springer-Verlag, New York, 1984, pp. 502-510.
- Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), no. 1, 63–69. MR 766095, DOI 10.1007/BF01890022
- Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1 (1985), no. 3, 231–248. MR 891530, DOI 10.1007/BF01890033
- Attila Máté, Paul Nevai, and Vilmos Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), no. 2, 239–281. MR 882423, DOI 10.2307/2374574 —, Extensions of Szegö’s theory of orthogonal polynomials. II, Constructive Approximation 3 (1987). —, Extensions of Szegö’s theory of orthogonal polynomials. III, Constr. Approx. 3 (1987). —, Twisted difference operators and perturbed Chebyshev polynomials, manuscript.
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Paul G. Nevai, On orthogonal polynomials, J. Approx. Theory 25 (1979), no. 1, 34–37. MR 526275, DOI 10.1016/0021-9045(79)90031-5
- Paul G. Nevai, An asymptotic formula for the derivatives of orthogonal polynomials, SIAM J. Math. Anal. 10 (1979), no. 3, 472–477. MR 529065, DOI 10.1137/0510044
- Paul G. Nevai, Orthogonal polynomials defined by a recurrence relation, Trans. Amer. Math. Soc. 250 (1979), 369–384. MR 530062, DOI 10.1090/S0002-9947-1979-0530062-6
- Paul Nevai, Two of my favorite ways of obtaining asymptotics for orthogonal polynomials, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 417–436. MR 820541 —, Extensions of Szegö’s theory of orthogonal polynomials, Orthogonal Polynomials and Their Applications (C. Brezinski et al., eds.), Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1985, pp. 230-238. —, "Géza Freud, orthogonal polynomials and Christoffel functions, J. Approximation Theory 48 (1986). P. Nevai and V. Totik, Orthogonal polynomials and their zeros, manuscript.
- E. M. Nikishin, An estimate for orthogonal polynomials, Acta Sci. Math. (Szeged) 48 (1985), no. 1-4, 395–399 (Russian). MR 810895 E. A. Rahmanov, On the asymptotics of the ratio of orthogonal poynomials, Math. USSR Sb. 32 (1977), 199-213. —, On the asymptotics of the ratio of orthogonal polynomials. II, Math. USSR Sb. 46 (1983), 105-117.
- J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. I, American Mathematical Society, New York, 1943. MR 0008438 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939, 4th ed., 1975.
Bibliographic Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 300 (1987), 175-189
- MSC: Primary 42C05
- DOI: https://doi.org/10.1090/S0002-9947-1987-0871671-9
- MathSciNet review: 871671