A survey of optimal polynomial approximants, applications to digital filter design, and related open problems

TL;DR

This survey explores optimal polynomial approximants in Hardy spaces, their mathematical properties, applications in digital filter design, and highlights open problems in the field.

Contribution

It consolidates known results on optimal polynomial approximants, their connections to orthogonal polynomials, and discusses open questions in the area.

Findings

Optimal polynomial approximants are linked to orthogonal polynomials and reproducing kernels.

Zeros of optimal polynomial approximants have specific distribution properties.

Convergence rates and decay behaviors are characterized for these approximants.

Abstract

In the last few years, the notion of optimal polynomial approximant has appeared in the mathematics literature in connection with Hilbert spaces of analytic functions of one or more variables. In the 70s, researchers in engineering and applied mathematics introduced least-squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space $H^2$ these objects are identical. This paper is a survey of known results about optimal polynomial approximants. In particular, we will examine their connections with orthogonal polynomials and reproducing kernels in weighted spaces and digital filter design. We will also describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results. Throughout the paper, we state many open questions that may be of interest.