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Verblunsky coefficients and Nehari sequences

Authors: Yukio Kasahara and Nicholas H. Bingham
Journal: Trans. Amer. Math. Soc. 366 (2014), 1363-1378
MSC (2010): Primary 42C05; Secondary 42A10, 42A70
Published electronically: July 18, 2013
MathSciNet review: 3145734
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Abstract: We are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle. In general, the Szegö function is determined by its modulus, while the condition in question is that it is also determined by its argument, or in terms of the function theory, that the square of the Szegö function is rigid. In prediction theory, this is known as a spectral characterization of complete nondeterminacy for stationary processes, studied by Bloomfield, Jewel and Hayashi (1983) going back to a small but important result in the work of Levinson and McKean (1964). It is also related to the cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem, and there is a one-to-one correspondence between the Verblunsky coefficients and the negatively indexed Fourier coefficients of the phase factor of the Szegö function, which we call a Nehari sequence. We present some fundamental results on the correspondence, including extensions of the strong Szegö and Baxter's theorems.

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  • [AAK] V. M. Adamjan, D. Z. Arov and M. G. Krein, Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and I. Schur, Funkcional. Anal. i Priložen. 2 (1968), 1-17; Functional Anal. Appl. 2 (1968), 269-281. MR 0636333 (58:30446)
  • [B] G. Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471-487. MR 0126126 (23:A3422)
  • [BIK] N. H. Bingham, A. Inoue and Y. Kasahara, An explicit representation of Verblunsky coefficients, Statist. Probab. Lett., 82 (2012), no. 2, 403-410. MR 2875229
  • [BJH] P. Bloomfield, N. P. Jewell and E. Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), 307-317. MR 705750 (85g:60053)
  • [D] P. L. Duren, Theory of $ H\sp {p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [DM] H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Academic Press, New York, 1976. MR 0448523 (56:6829)
  • [G] J. B. Garnett, Bounded analytic functions. Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [GI] B. L. Golinskii and I. A. Ibragimov, A limit theorem of G. Szegö, Izv. Akad. Nauk SSSR Ser. Mat 35 (1971), 408-427, [Russian] MR 0291713 (45:804)
  • [H] P. R. Halmos, A Hilbert space problem book. Springer-Verlag, New York, 1982. MR 675952 (84e:47001)
  • [Ha] E. Hayashi, The solution sets of extremal problems in $ H^1$, Proc. Amer. Math. Soc. 93 (1985), 690-696. MR 776204 (86e:30035)
  • [Ho] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, 1962 MR 0133008 (24:A2844)
  • [I] I. A. Ibragimov, A theorem of Gabor Szegö, Mat. Zametki 3 (1968), 693-702 [Russian]. MR 0231114 (37:6669)
  • [IS] I. A. Ibragimov and V. N. Solev, A condition for the regularity of a Gaussian stationary process, Dokl. Akad. Nauk SSSR 185 (1969), 509-512 [Russian]. English transl. in Soviet Math. Dokl. 10 (1969), 371-375. MR 0242243 (39:3576)
  • [In1] A. Inoue, Asymptotics for the partial autocorrelation function of a stationary process, J. Anal. Math. 81 (2000), 65-109. MR 1785278 (2001e:60075)
  • [In2] A. Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes, Ann. Appl. Probab. 12 (2002), 1471-1491. MR 1936600 (2003i:62143)
  • [In3] A. Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), 523-551. MR 2365483 (2009b:62176)
  • [IK1] A. Inoue and Y.  Kasahara, Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing, J. Multivariate Anal. 89 (2004), 135-147. MR 2041213 (2005a:62197)
  • [IK2] A. Inoue and Y.  Kasahara, Explicit representation of finite predictor coefficients and its applications, Ann. Statist. 34 (2006), no. 2, 973-993. MR 2283400 (2008c:60036)
  • [LM] N. Levinson and H. P. McKean, Weighted trigonometrical approximation on $ R\sp {1}$ with application to the germ field of a stationary Gaussian noise. Acta Math. 112 (1964), 99-143. MR 0163111 (29:414)
  • [dLR] K. de Leeuw and W. Rudin, Extreme points and extremum problems in $ H\sb {1}$, Pacific J. Math. 8 (1958), 467-485. MR 0098981 (20:5426)
  • [N1] T. Nakazi, Exposed points and extremal problems in $ H^1$, J. Funct. Anal. 53 (1983), 224-230. MR 724027 (85f:46096)
  • [N2] T. Nakazi, Kernels of Toeplitz operators, J. Math. Soc. Japan 38 (1986), 607-616. MR 856129 (87k:47061)
  • [Ne] Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153-162. MR 0082945 (18:633f)
  • [P] V. V. Peller, Hankel operators and their applications. Springer-Verlag, New York, 2003. MR 1949210 (2004e:47040)
  • [PS] A. Poltoratski and D. Sarason, Aleksandrov-Clark measures, Recent advances in operator-related function theory, 1-14, Contemp. Math. 393, Amer. Math. Soc., Providence, RI, 2006. MR 2198367 (2006i:30048)
  • [S1] D. Sarason, Function theory on the unit circle, Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, 1978. MR 521811 (80d:30035)
  • [S2] D. Sarason, Exposed points in $ H\sp 1$, I, Oper. Theory Adv. Appl. 41 (1989), 485-496. MR 1038352 (91h:46043)
  • [S3] D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, Wiley, 1994. MR 1289670 (96k:46039)
  • [Se] A. Seghier, Prédiction d'un processus stationnaire du second ordre de covariance connue sur un intervalle fini, Illinois J. Math. 22 (1978), no. 3, 389-401. MR 497482 (80j:60060)
  • [Si1] B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, American Mathematical Society, Providence, RI, 2005. MR 2105088 (2006a:42002a)
  • [Si2] B. Simon, Orthogonal polynomials on the unit circle, Part 2. Spectral theory, American Mathematical Society, Providence, RI, 2005. MR 2105089 (2006a:42002b)
  • [Si3] B. Simon, Szegö's theorem and its descendants. Spectral theory for $ L^2$ perturbations of orthogonal polynomials, Princeton University Press, Princeton, NJ, 2011. MR 2743058 (2012b:47080)
  • [Sz1] G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939
  • [Sz2] G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Sém. Math. Univ. Lund 1952 (1952), Tome Supplementaire, 228-238. MR 0051961 (14:553d)
  • [V1] S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. 38 (1935), 125-157.
  • [V2] S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. 40 (1936), 290-320.
  • [Y] K. Yabuta, Some uniqueness theorems for $ H^{p}(U^{n})$ functions. Tôhoku Math. J. 24 (1972), 353-357. MR 0340647 (49:5399)
  • [Yo] R. Younis, Hankel operators and extremal problems in $ H^1$, Integral Equations Operator Theory 9 (1986), 893-904. MR 866970 (88c:47049)

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Additional Information

Yukio Kasahara
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Nicholas H. Bingham
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ United Kingdom

Keywords: Orthogonal polynomials on the unit circle, Verblunky coefficients, Nehari problem, rigid functions
Received by editor(s): December 3, 2011
Published electronically: July 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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