Author's School

Arts & Sciences

Author's Department

Mathematics

Document Type

Article

Publication Date

5-1-2016

Originally Published In

Journal of Functional Analysis, Volume 270, Issue 9, 1 May 2016, Pages 3505–3558. doi:10.1016/j.jfa.2016.02.002

Abstract

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the “split-shift orthogonality condition” and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This approach allows several refinements of the characterization and it also allows us to prove a sums of squares decomposition which at once generalizes the Cole–Wermer sums of squares result for two variable stable polynomials as well as a sums of squares result related to the Schur–Cohn method for counting the roots of a univariate polynomial in the unit disk.

Comments

This is an author manuscript version of article published in Journal of Functional Analysis, Volume 270, Issue 9, 1 May 2016, Pages 3505. doi: 10.1016/j.jfa.2016.02.002 © 2016 Elsevier

DOI

10.1016/j.jfa.2016.02.002

Embargo Period

5-1-2018

Available for download on Tuesday, May 01, 2018

Included in

Mathematics Commons

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