1,009 research outputs found
Polynomials with no zeros on the bidisk
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.Comment: 52 page
Nevanlinna-Pick interpolation on distinguished varieties in the bidisk
This article treats Nevanlinna-Pick interpolation in the setting of a special
class of algebraic curves called distinguished varieties. An interpolation
theorem, along with additional operator theoretic results, is given using a
family of reproducing kernels naturally associated to the variety. The examples
of the Neil parabola and doubly connected domains are discussed.Comment: 31 pages. The question left open at the end of version 1 has been
answered in the affirmative; see Theorem 1.12 and Corollary 1.13 in version
Realization of functions on the symmetrized bidisc
We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidisc G = def {( z + w , z w ) : | z | < 1 , | w | < 1} . As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions
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