Mike Jury: An Optimal Approximation Problem for Free Polynomials

Abstract:

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial \(f\) in \(d\) noncommuting arguments, find an nc polynomial \(p_n\), of degree at most \(n\), to minimize \(c_n := \|p_nf − 1\|^2\). (Here the norm is the \(\ell^2\) norm on coefficients.) We show that \(c_n\to 0\) if and only if \(f\) is nonsingular in a certain noncommutative domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the \(d\)-shift. (This is joint work with P. Arora, M. Augat, and M. Sargent.)