A survey of optimal polynomial approximants and connections to digital filters

Abstract: In this talk, I will discuss the notion of optimal polynomial approximants, which are polynomials that approximate, in some sense, inverses of functions in certain Hilbert spaces of analytic functions. In the last 10 years, a number of papers have appeared examining the zeros of these polynomials, rates of convergence, efficient algorithms for their computation, and connections to orthogonal polynomials and reproducing kernels, among other topics. On the other hand, 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 $H^2$ the optimal polynomial approximants and the least squares inverses are identical. In this talk, I will survey results related to the zeros of optimal polynomial approximants and implications for the design of ideal digital filters. This talk is based on a preprint of a survey paper that is joint with Ray Centner.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

Series comments: Please e-mail R.Halburd@ucl.ac.uk for the Zoom link. Also, please let me know whether you would like to be added to the mailing list to automatically receive links for future talks in CAvid.