Lower bounds in $H^2$-rational approximation to Blaschke products

TL;DR

This paper establishes lower bounds for the best rational approximation in the Hardy space to finite Blaschke products, focusing on specific cases like powers of z and more general products with zeros away from the boundary.

Contribution

It introduces new lower bounds for rational approximation in H^2 to Blaschke products, including Fourier coefficient estimates for these functions.

Findings

Lower bounds for approximation to z^N in H^2.

Extension of bounds to general Blaschke products with zeros bounded away from the circle.

Development of Fourier coefficient estimates for Blaschke products.

Abstract

We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.