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Abstract
We prove a detailed sums of squares formula for two-variable polynomials with no zeros on
the bidisk
D 2 ,
extending previous such formulas by Cole and Wermer and by Geronimo and Woerdeman.
Our 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 Fejér–Riesz factorizations, analytic extension theorems for a class
of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.
Keywords
bidisk, Christoffel–Darboux, sums of squares, Fejér–Riesz,
orthogonal polynomials, distinguished varieties, Pick
interpolation, Andô's inequality, Bernstein–Szegő measures,
torus, stable polynomials
Mathematical Subject Classification 2000
Primary: 42C05
Secondary: 47A57, 46C07, 42B05, 14M12
Milestones
Received: 23 October 2008
Revised: 20 October 2009
Accepted: 3 December 2009
Published: 8 June 2010