The classical Fej\\‘er-Riesz Theorem states that a nonnegative trigonometric polynomial can be factored as the absolute square of an analytic polynomial. Indeed, the factorization can be done with an outer polynomial. Various generalizations of this result have been considered. For example, Rosenblum showed that the result remained true for operator valued trigonometric polynomials. If one instead considers operator valued polynomials in several variables, one obtains factorization results for strictly positive polynomials, though outer factorizations become much more problematic. In another direction, Scott McCullough proved a factorization result for so-called hereditary trigonometric polynomials in freely noncommuting variables (strict positivity not needed). In this talk we consider an analogue of (hereditary) trigonometric polynomials over discrete groups, and give a result which includes a strict form of McCullough’s theorem as well as the multivariable version of Rosenblum’s theorem.

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Last updated: 11 Dec 2023