Verblunsky coefficients and Nehari sequences

We are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle. In general, the Szego 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 Szego 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 with the cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem, and there is a one-one correspondence between the Verblunsky coefficients and the negatively indexed Fourier coefficients of the phase factor of the Szego function, which we call a Nehari sequence. We presents some fundamental results on the correspondence, including extensions of the strong Szego and Baxter’s theorems.

[1]  Russell S. Peak,et al.  Part 2: , 2020, Journal of Neural Transmission.

[2]  N. Bingham,et al.  An explicit representation of Verblunsky coefficients , 2011, 1109.4513.

[3]  ICOLAS,et al.  CHARACTERIZATIONS OF COMPLETELY NONDETERMINISTIC STOCHASTIC PROCESSES , 2012 .

[4]  H. Hakimzadeh,et al.  Part 1 , 2011 .

[5]  Herbert K. H. Lee,et al.  Gaussian Processes , 2011, International Encyclopedia of Statistical Science.

[6]  B. Simon.  Szegő's Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials , 2010 .

[7]  E. Hayashi.  THE SOLUTION SETS OF EXTREMAL PROBLEMS IN H 1 , 2010 .

[8]  A. Inoue.  AR and MA representation of partial autocorrelation functions, with applications , 2007, math/0702648.

[9]  V. Peller.  Hankel Operators and Their Applications , 2003, IEEE Transactions on Automatic Control.

[10]  Rene F. Swarttouw,et al.  Orthogonal Polynomials , 2005, Series and Products in the Development of Mathematics.

[11]  B. Simon.  Rakhmanov’s theorem and related issues , 2005 .

[12]  Barry Simon,et al.  Orthogonal polynomials on the unit circle. Part 1 , 2005 .

[13]  Akihiko Inoue,et al.  Explicit representation of finite predictor coefficients and its applications , 2004, math/0405051.

[14]  A. Inoue,et al.  Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing , 2000 .

[15]  A. Inoue,et al.  Asymptotic behaviour for partial autocorrelation functions of fractional ARIMA processes , 2000 .

[16]  P. Duren.  Theory of Hp Spaces , 2000 .

[17]  A. Inoue.  Asymptotics for the partial autocorrelation function of a stationary process , 2000 .

[18]  M. Nimchek.  Banach spaces of analytic functions , 1996 .

[19]  D. Sarason.  Sub-Hardy Hilbert Spaces in the Unit Disk , 1994 .

[20]  D. Sarason.  Kernels of Toeplitz Operators , 1994 .

[21]  J. Conway.  Function theory on the unit circle , 1991 .

[22]  D. Sarason.  Exposed Points in H 1, I , 1989 .

[23]  R. Younis.  Hankel operators and extremal problems in H1 , 1986 .

[24]  E. Hayashi.  The solution sets of extremal problems in , 1985 .

[25]  Exposed points and extremal problems in H1 , 1983 .

[26]  G. Szegö,et al.  [52–2] On Certain Hermitian Forms Associated with the Fourier Series of a Positive Function , 1982 .

[27]  A. Seghier.  Prédiction d'un processus stationnaire du second ordre de covariance connue sur un intervalle fini , 1978 .

[28]  D. Sarason.  Function theory on the unit circle , 1978 .

[29]  Harry Dym,et al.  Gaussian processes, function theory, and the inverse spectral problem , 1976 .

[30]  SOME UNIQUENESS THEOREMS FOR {H^p}({U^n}) FUNCTIONS , 1972 .

[31]  O. H. Lowry.  Academic press. , 1972, Analytical chemistry.

[32]  P. Duren.  Theory of H[p] spaces , 1970 .

[33]  F. Smithies.  A HILBERT SPACE PROBLEM BOOK , 1968 .

[34]  A. Devinatz.  An extension of a limit theorem of G. Szegö , 1966 .

[35]  N. Levinson,et al.  Weighted trigonometrical approximation onR1 with application to the Germ field of a stationary Gaussian noise , 1964 .

[36]  G. Baxter,et al.  A CONVERGENCE EQUIVALENCE RELATED TO POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE , 1961 .

[37]  W. Rudin,et al.  Extreme points and extremum problems in H1 , 1958 .

[38]  Z. Nehari.  On Bounded Bilinear Forms , 1957 .

[39]  Function Theory , 1951, Nature.

[40]  Z. Nehari.  Bounded analytic functions , 1950 .

[41]  S. Verblunsky,et al.  On Positive Harmonic Functions , 1936 .

[42]  S. Verblunsky,et al.  On Positive Harmonic Functions: A Contribution to the Algebra of Fourier Series , 1935 .