  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 $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as
a power of $n$. This is motivated by Sch\"affer's question on norms of
inverses.
