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 f in d d freely noncommuting arguments, find a free polynomial p n p_n , of degree at most n n , to minimize c n ≔ ‖ p n f − 1 ‖ 2 c_n ≔\|p_nf-1\|^2 . (Here the norm is the ℓ 2 \ell ^2 norm on coefficients.) We show that c n → 0 c_n\to 0 if and only if f f is nonsingular in a certain nc 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 d -shift.