On the Koebe Quarter Theorem for Polynomials

Note on a Theorem of Bôcher and Koebe*

A Theorem on Difference Polynomials

We shall prove the following analogue of a theorem of Kronecker's : Let J be a difference field containing an element t which is distinct from its transforms of any order. Every perfect ideal in the ring 7\yu ' ' ' y y<n\ has a basis consisting of n + 1 difference polynomials. The corresponding theorem for differential polynomials was proved by J. F. Ritt. We shall follow in all but details a b...

On the continuous analog of Rakhmanov's theorem for orthogonal polynomials

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov’s theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle. 1. Rakhmanov’s t...

Aizenman’s Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if E (∫ dθ 2π ∣∣∣ ( C + eiθ C − eiθ ) k ∣∣∣ p) ≤ C1e−κ1|k− | for some κ1 > 0 and p < 1, then for suitable C2 and κ2 > 0, E ( sup n |(C)k | ) ≤ C2e−κ2|k− |. Here C is the CMV matrix.