On Szegő's theorem for a nonclassical case

Abstract

In this paper we prove Szegő's Theorem for the case when a finite number of Verblunsky coefficients lie outside the closed unit disk. Although a form of this result was already proved by A.L. Sakhnovich, we use a very different method, which shows that the OPUC machinery can still be applied to deal with such nonclassical cases. The basic tool we use is Khrushchev's formula that in the classical case relates the absolutely continuous part of the measure and the N -th iterate of the Schur algorithm. It is noteworthy that Khrushchev's formula makes the proof short and extremely transparent. Also, we discuss Verblunsky's theorem for the case in question.

Asymptotics for polynomials orthogonal in an indefinite metric

“…To formulate the next result, which is a generalization of [15, Theorem 2.4.1, part (iv)], we should recall from [4] that the zeros of ϕ * n inside D will either tend to the poles of F in D or to the boundary of the unit disk as n → ∞. So, let B n be the Blaschke product formed by the zeroes of ϕ * n inside D, that is…”

“…In [4] an analog of Szegő's Theorem was proven in this non-classical setting. The aim of the present paper is to further develop the asymptotic theory of polynomials generated from the Szegő recursion using a sequence satisfying (1.1).…”

“…We can then mimic the proof of [16, Theorem 13.1.2] to reach the desired conclusion in some neighborhood of zero. We know from [4] that F is holomorphic in D away from a finite set of isolated singularities, so the relation (3.8) provides an analytic continuation of m(z + z −1 ) to the domain of F and shows that (3.8) holds there. Finally, [4,Lemma 3.4] implies that the poles of F in D coincide with the zeroes of B.…”

“…In [4], a special non-classical class of Verblunsky coefficients was studied and unlike the classical case, that class does not correspond to meausres on the unit circle T. In this note we proceed with the exploration of this class and we will use the notation from [4], which is in turn inherited from [15,16]. More precisely, we consider sequences {α n } ∞ n=0 of complex numbers for which there exists a natural number N such that |α n | = 1, n = 0, 1, 2, .…”

Jacobi matrices generated by ratios of hypergeometric functions

“…⊤ , which is proved to be a generalized Nevanlinna function. Remarkably, some nonclassical orthogonal polynomials on the unit disk were introduced in [7] and the Szegő mapping applied to those polynomials leads to tridiagonal matrices that have the same structure as H does [8].…”