Approximation by polynomials and Blaschke products having all zeros on a circle

TL;DR

This paper demonstrates how nonvanishing analytic functions can be approximated by Blaschke products with zeros on a specific circle, providing new proofs and connecting to Riemann zeta function universality.

Contribution

It introduces a novel approximation method using Blaschke products with zeros on a circle and offers a new proof for polynomial approximation results.

Findings

Approximation of nonvanishing analytic functions by Blaschke products with zeros on a circle

New proof of classical polynomial approximation result

Connection established between approximation and Riemann zeta function universality

Abstract

We show that a nonvanishing analytic function on a domain in the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the domain. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.