Polynomials with no zeros on a face of the bidisk

详细信息    查看全文

• 作者：Jeffrey S. Geronimo^a ; ¹ ; ^{geronimo@math.gatech.edu" class="auth_mail" title="E-mail the corresponding author} ; Plamen Iliev^a ; ² ; ^{iliev@math.gatech.edu" class="auth_mail" title="E-mail the corresponding author} ; Greg Knese^b ; ³ ; ^{geknese@math.wustl.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：42C05 ; 47A57 ; 30E05
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：270
• 期：9
• 页码：3505-3558
• 全文大小：696 K

文摘

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.