Verblunsky coefficients and Nehari sequences

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.

Matricial Baxter's theorem with a Nehari sequence

“…Since , by von Neumann's alternating projections, converges to the orthogonal projection onto their intersection as . Henceforth, when is rigid, it follows from that In fact, when is rigid, coincides with the space of ‐valued polynomials with degree less than n for , and analogous projection formulas can be used for studying MOPUC, as in the authors' earlier work ; see also Inoue–Kasahara–Pourahmadi for applications to prediction problems.…”

“…The following fact (cf. [, Remark 5.2]) will be helpful to understand the argument in the proof of Lemma below. Proposition Assume .…”

“…In fact, when ℎ ℎ is rigid, 2 ∩ 2 coincides with the space of -valued polynomials with degree less than for = 0, 1, 2 …, and analogous projection formulas can be used for studying MOPUC, as in the authors' earlier work [13]; see also Inoue-Kasahara-Pourahmadi [12] for applications to prediction problems.…”

“…The main contribution of this note is to give an additional condition (v), motivated by the authors' previous result [, Theorem 5.8], to the known equivalence among (i)–(iv) in the following Baxter's theorem. Theorem Let be the Verblunsky coefficients of a non‐trivial ‐valued measure μ corresponding to the Carathéodory function F and Schur function f .…”

“…Recently, based on the investigation for the Nehari problem by Adamjan-Arov-Kreȋn [2] and Adamjan [1], the authors [13,14] pointed out that matrix measures (with some constraint) are also parametrized by alternative sequences = ( 1 , 2 , …) of strictly contractive matrices (with some constraint), called Nehari sequences, and established some basic results on the correspondence among , and . This note presents Baxter's theorem in terms of a Nehari sequence, namely, has the properties just mentioned if and only if is summable, answering an open question posed by the second author [6].…”