Zeros of Non-Baxter Paraorthogonal Polynomials on the Unit Circle

Abstract

Abstract. We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (−1, 0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.

On the spacing of zeros of paraorthogonal polynomials for singular measures

“…For random Verblunsky coefficients, the works [11,27] show a transition from Poisson to clock behavior via asymptotic β-ensemble statistics (indeed showing, in this particular case, a correlation between measure continuity and local repulsion). From a slightly different perspective, the papers [7,13,20,22] study the connection between regularity properties of {α n } ∞ n=0 and these spacings. In particular [13] obtain sufficient conditions ensuring clock behavior, whereas [7,22] obtain global upper bounds on the spacing depending on the decay rate of {α n } ∞ n=0 .…”

“…In fact, they are eigenvalues of a unitary truncation of the CMV matrix associated with the Verblunsky coefficients {α n } ∞ n=0 in much the same way as the zeros of the n'th OPRL are the eigenvalues of a self-adjoint truncation of a Jacobi matrix (see [26, Sections 2.2 and 8.2] for details). Other relevant references include [4,5,17,20,23,24,29]. Questions about the asymptotic distribution of these zeros on ∂D are thus natural and have been studied in various contexts which we discuss in greater detail below.…”

Zero Spacings of Paraorthogonal Polynomials on the Unit Circle

“…We will be interested in properties of the zeros of these polynomials and there exists a substantial literature on that subject. Many results have been proven about interlacing properties (see [1,19,23]), zero spacing (see [4,18,9,14]), and the relationship between the zeros and the underlying measure of orthogonality (see [11,19]). Our primary interest will be in the spacing between zeros, though we will also have something to say about the bulk distribution of zeros.…”

“…Similar objects have also been studied in [8,9,10,14]. Though our definition only defines η n up to an integer multiple of 2π, we will be looking at changes in η n (θ) as θ changes, so our choice of normalization will be irrelevant.…”

An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

“…Paraorthogonal polynomials and their zeros have received considerable recent attention from the research community (see for example [2,8,17,18,19,27,29,35,37]). It is wellknown and easy to show that all of the zeros of Φ n (z; β; µ) are distinct and lie on the unit circle.…”