An Optimal Approximation Problem For Free Polynomials

2022/09/21 by Palak Arora, Meric Augat, Arora, Palak +5 · 1 citation
Mathematics · #Meromorphic and Entire Functions #Analytic and geometric function theory #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2209.10373

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 freely noncommuting arguments, find a free polynomial p_n, of degree at most n, to minimize c_n := ‖p_nf-1‖². (Here the norm is the ℓ² norm on coefficients.) We show that c_n→ 0 if and only if 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-shift.

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• Metric projections, zeros of optimal polynomial approximants, and some extremal problems in Hardy spaces

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