Polynomials with no zeros on the bidisk

Abstract: We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk extending previous versions of such a formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to the Christoffel-Darboux formula for orthogonal polynomials on the unit circle, but the extension to two variables involves issues of uniqueness in the formula and the study of ideals of two variable orthogonal polynomials with respect to a positive Borel measure on the torus which may have infinite mass. We present applications to two variable Fej´er-Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.

Polynomials defining distinguished varieties.  (submitted for publication)

Abstract: Using a sums of squares formula for two variable
polynomials with no zeros on the bidisk, we are able to give a new
proof of a representation formula for distinguished varieties.  For
distinguished varieties with no singularities on the two-torus, we
are able to provide extra details about the representation formula
and use this to prove a bounded extension theorem.

Bernstein-Szeg\”o measures on the two dimensional torus, Indiana Univ. Math. J. 57 No. 3 (2008), 1353–1376

This paper can also be found here.

Abstract:

We present a new viewpoint (namely, reproducing kernels) and new proofs for several recent results of J. Geronimo and H. Woerdeman on orthogonal polynomials on the two dimenional torus (and related subjects). In addition, we show how their results give a new proof of Ando’s inequality via an equivalent version proven by Cole and Wermer. A simple necessary and sufficient condition for two variable polynomial stability is also given.

Update: In case anyone wants to use the symbols in the above paper, here is the metafont file, the style file, and a description of the symbols and commands (and how to use them).

Metafont file

Style file

Description of the gksymb package

The paper “Function theory on the Neil parabola” has appeared in print. Here is a link to the project euclid page for it:

Function theory on the Neil parabola

Better yet, you can find it here.  It should be cited something like:

Greg Knese, Function theory on the Neil parabola, Michigan Math. J., Vol. 55, Issue 1 (2007), 139-154.

Abstract: In this paper, we compute the Caratheodory distance on the Neil parabola, namely the set , making it the first nontrivial computation of this important (biholomorphically) invariant metric for a variety with a singularity. In addition, we compute the Caratheodory metric for the Neil parabola, relate the above computations to a certain bounded analytic interpolation problem for which known theorems do not apply, and present a bounded analytic extension result for functions on the Neil parabola to the bidisk.

Update:

Some of the formulas in this paper have been generalized to varieties of the form in the preprint:

Invariant metrics and distances on generalized Neil parabolas

Authors: Nikolai Nikolov, Peter Pflug

My paper “A Schwarz lemma on the polydisk” has “officially” appeared on the Proceedings of the AMS website.

A Schwarz lemma on the polydisk

This paper can also be found here.  For future reference the paper is cited as:

Greg Knese, A Schwarz lemma on the polydisk, Proc. Amer. Math. Soc., 135 (2007), 2759-2768

Abstract: This paper concerns a generalization of the infinitesimal portion of the classical Schwarz lemma inequality to the setting of the polydisk. Specifically, we give a complete description of the functions that are extremal for this inequality at every point of the polydisk: they are the transfer function of a symmetric unitary. In addition, some sufficient conditions are given for a function to be of this type.

Update: This paper has been referenced in recent preprint of J.M. Anderson, M.A. Dritschel, and J. Rovnyak. It can be found here.

Update 2: At the end of the paper we mention that we aren’t sure whether the extremal functions in the this paper always have the property that they cannot be continuously extended to the closed polydisk (except in the case of one variable). I realized recently that they indeed cannot be extended to the closed polydisk for the following reason. Rational inner functions on the polydisk which are continuous up to the boundary must be a Blaschke product of a fixed degree on every balanced disk. The extremal functions of this paper have the special property that through every point there is a balanced disk on which they are a Blaschke product of degree ONE! This is not true for every balanced disk, and therefore they cannot be extended continuously to the boundary. This all follows from the fact that continuity on the boundary forces the winding number of the holomorphic function on every balanced disk to be constant. I’m leaving out some details and being a little vague, but this is the idea.