Factoring non-negative operator valued trigonometric polynomials in two variables

Michael Dritschel (Newcastle University)

Thursday 21st March 16:00-17:00 Maths 311B

Abstract

The venerable Fejér-Riesz theorem, on the factorization of a non-negative complex valued trigonometric polynomial in one variable as the hermitian square of an analytic polynomial, is an essential tool in both pure mathematics and engineering, especially in signal processing.  There have been numerous generalizations, along with a keen interest in finding an analogue in two variables.  There are results for strictly positive polynomials in any number of variables, but they do not extend to non-negative polynomials, and it is known that such a factorization for non-negative polynomials in three or more variables does not hold.  In this talk, a generalization to operator valued non-negative trigonometric polynomials in two variables - factored as a finite sum of hermitian squares of analytic polynomials with tight control over the number and degrees of the analytic polynomials - is discussed.  The proof involves Schur complement techniques, along with a novel application of ultraproducts reminiscent of the Tarski transfer principle in real algebra.  While this is apparently non-constructive, it nevertheless leads to a concrete algorithm for factorization.

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