Optimal polynomial approximants

The notion of optimal polynomial approximants arises naturally from classical problems in function theory. Such polynomials approximate, in some optimal sense, reciprocals of functions in certain spaces on the unit disc, including the Hardy, Bergman and Dirichlet spaces. This talk gives an overview of the subject, focusing on the limiting behaviour of optimal polynomial approximants on subsets of the unit circle, and discusses a related problem on zero-free approximation of independent interest.

Further reading: Bénéteau–Centner, “A survey of optimal polynomial approximants, applications to digital filter design, and related open problems” (arXiv:2102.01725).