Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for publication in Adv. Mat

Fine Structure of the Zeros of Orthogonal Polynomials: A Review

We review recent work on zeros of orthogonal polynomials

Geometric Properties of Partial Sums of Univalent Functions

The $n$th partial sum of an analytic function $f(z)=z+\sum_{k=2}^\infty a_k z^k$ is the polynomial $f_n(z):=z+\sum_{k=2}^n a_k z^k$. A survey of the univalence and other geometric properties of the $n$th partial sum of univalent functions as well as other related functions including those of starlike, convex and close-to-convex functions are presented

Distributions on unbounded moment spaces and random moment sequences

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.Comment: Published in at http://dx.doi.org/10.1214/11-AOP693 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spectral transformations for Hermitian Toeplitz matrices

22 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2319946 (2008c:42025)Zbl#: Zbl 1131.42301In this paper we deal with some perturbations of probability measures supported on the unit circle as well as, in a more general framework, with Hermitian linear functionals. We focus our attention in the Hessenberg matrix associated with the multiplication operator in terms of an orthogonal basis in the linear space of polynomials with complex coefficients. The LU and QR factorizations of such a matrix are introduced. Then, the connection between the above-mentioned perturbations and such factorizations is presented.The work of the first author (Leyla Daruis) was supported by Ministerio de Educación y Ciencia of Spain, under Grant MTM 2005-08571. The work of the of second author (Javier Hernández) was supported by Fundación Universidad Carlos III de Madrid. The work of the third author (Francisco Marcellán) was supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, under Grant BFM 2003-06335-C03-02, and INTAS Research Network NeCCA INTAS 03-51-6637.Publicad

Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy

We provide a detailed derivation of all complex-valued algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy. In addition, we survey a recursive construction of the Ablowitz-Ladik hierarchy and its zero-curvature and Lax formalism.Comment: 41 page