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Matrix Completions, Moments, and Sums of Hermitian Squares$
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Mihály Bakonyi and Hugo J. Woerdeman

Print publication date: 2011

Print ISBN-13: 9780691128894

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691128894.001.0001

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Multivariable moments and sums of Hermitian squares

Multivariable moments and sums of Hermitian squares

(p.175) Chapter Three Multivariable moments and sums of Hermitian squares
Matrix Completions, Moments, and Sums of Hermitian Squares

Mihály Bakonyi

Hugo J. Woerdeman

Princeton University Press

This chapter shows how five closely related problems become vastly different when one considers the multivariable case. In a similar way, various natural sums of Hermitian squares problems that one may consider in the multivariable case all come down to the classical factorization problem in one variable as discussed in Section 1.1. The discussions cover positive Carathéodory interpolation on the polydisk; inverses of multivariable Toeplitz matrices and Christoel–Darboux formulas; two-variable moment problem for Bernstein–Szeg ő measures; Fejéer–Riesz factorization and sums of Hermitian squares; completion problems for positive semidefinite functions on amenable groups; moment problems on free groups, noncommutative factorization; two-variable Hamburger moment problem; and Bochner's theorem and an application to autoregressive stochastic processes. Exercises and notes are provided at the end of the chapter.

Keywords:   multivariables, Hermitian squares problems, positive Carathéodory interpolation, Toeplitz matrices, Christoel–Darboux formulas, Bernstein–Szeg ő measures, Fejéer–Riesz factorization

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