On the Fourier coefficients of powers of a finite Blaschke product

Abstract

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the ${\ell }^{\infty }$-norm of the sequence of the Fourier coefficients of $B^n$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det T|\cdot \Vert T^{-1}\Vert \cdot \Vert T{\Vert }^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses.