On the Fourier coefficients of powers of a finite Blaschke product

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the ${\mathrm{\ell }}^{\mathrm{\infty }}$-norm of the sequence of the Fourier coefficients of $B^n$ as $n$ tends to $\mathrm{\infty }$. 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 $\mathrm{\mid }\det T\mathrm{\mid }\cdot \mathrm{\parallel }{T}^{-1}\mathrm{\parallel }\cdot \mathrm{\parallel }T{\mathrm{\parallel }}^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.