Verblunsky coefficients and Nehari sequences

Kasahara, Yukio; Bingham, Nicholas

doi:10.1090/S0002-9947-2013-05874-6

Verblunsky coefficients and Nehari sequences
HTML articles powered by AMS MathViewer

by Yukio Kasahara and Nicholas H. Bingham PDF

Trans. Amer. Math. Soc. 366 (2014), 1363-1378 Request permission

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.

References

V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel matrices and generalized Carathéodory-Fejér and I. Schur problems, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 1–17 (Russian). MR 0636333
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
N. H. Bingham, Akihiko Inoue, and Yukio Kasahara, An explicit representation of Verblunsky coefficients, Statist. Probab. Lett. 82 (2012), no. 2, 403–410. MR 2875229, DOI 10.1016/j.spl.2011.11.004
Peter Bloomfield, Nicholas P. Jewell, and Eric Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), no. 2, 307–317. MR 705750
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Probability and Mathematical Statistics, Vol. 31, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0448523
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
B. L. Golinskiĭ and I. A. Ibragimov, A limit theorm of G. Szegő, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 408–427 (Russian). MR 0291713
Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
Eric Hayashi, The solution sets of extremal problems in $H^1$, Proc. Amer. Math. Soc. 93 (1985), no. 4, 690–696. MR 776204, DOI 10.1090/S0002-9939-1985-0776204-7
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
I. A. Ibragimov, A theorem of Gabor Szegő, Mat. Zametki 3 (1968), 693–702 (Russian). MR 231114
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). MR 0242243
Akihiko Inoue, Asymptotics for the partial autocorrelation function of a stationary process, J. Anal. Math. 81 (2000), 65–109. MR 1785278, DOI 10.1007/BF02788986
Akihiko Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes, Ann. Appl. Probab. 12 (2002), no. 4, 1471–1491. MR 1936600, DOI 10.1214/aoap/1037125870
Akihiko Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), no. 3-4, 523–551. MR 2365483, DOI 10.1007/s00440-007-0074-1
Akihiko Inoue and Yukio Kasahara, Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing, J. Multivariate Anal. 89 (2004), no. 1, 135–147. MR 2041213, DOI 10.1016/S0047-259X(02)00027-1
Akihiko Inoue and Yukio Kasahara, Explicit representation of finite predictor coefficients and its applications, Ann. Statist. 34 (2006), no. 2, 973–993. MR 2283400, DOI 10.1214/009053606000000209
N. Levinson and H. P. McKean Jr., Weighted trigonometrical approximation on $R^{1}$ with application to the germ field of a stationary Gaussian noise, Acta Math. 112 (1964), 99–143. MR 163111, DOI 10.1007/BF02391767
Karel de Leeuw and Walter Rudin, Extreme points and extremum problems in $H_{1}$, Pacific J. Math. 8 (1958), 467–485. MR 98981
Takahiko Nakazi, Exposed points and extremal problems in $H^{1}$, J. Funct. Anal. 53 (1983), no. 3, 224–230. MR 724027, DOI 10.1016/0022-1236(83)90032-0
Takahiko Nakazi, Kernels of Toeplitz operators, J. Math. Soc. Japan 38 (1986), no. 4, 607–616. MR 856129, DOI 10.2969/jmsj/03840607
Zeev Nehari, On bounded bilinear forms, Ann. of Math. (2) 65 (1957), 153–162. MR 82945, DOI 10.2307/1969670
Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
Alexei Poltoratski and Donald Sarason, Aleksandrov-Clark measures, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 1–14. MR 2198367, DOI 10.1090/conm/393/07366
Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR 521811
Donald Sarason, Exposed points in $H^1$. I, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988) Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 485–496. MR 1038352
Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1289670
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 (French). MR 497482
Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089, DOI 10.1090/coll/054.2/01
Barry Simon, Szegő’s theorem and its descendants, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. Spectral theory for $L^2$ perturbations of orthogonal polynomials. MR 2743058
G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939
G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplémentaire, 228–238. MR 51961
S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. 38 (1935), 125–157.
S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. 40 (1936), 290–320.
Kôzô Yabuta, Some uniqueness theorems for $H^{p}(U^{n})$ functions, Tohoku Math. J. (2) 24 (1972), 353–357. MR 340647, DOI 10.2748/tmj/1178241545
Rahman Younis, Hankel operators and extremal problems in $H^1$, Integral Equations Operator Theory 9 (1986), no. 6, 893–904. MR 866970, DOI 10.1007/BF01202522