On the Fourier Coefficients of Powers of a Finite Blaschke Product

Given a finite Blaschke product B we prove asymptotically sharp estimates on the ∞-norm of the sequence of the Fourier coefficients of Bⁿ as n tends to ∞. This norm decays as n-1/N for some N≥ 3. Furthermore, for every N≥ 3, we produce explicitly a finite Blaschke product B with decay n-1/N. As an application we construct a sequence of n× n invertible matrices T with arbitrary spectrum in the unit disk and such that the quantity |T|· \|T⁻¹\|· \|T\|¹⁻ⁿ grows as a power of n. This is motivated by Schäffer’s question on norms of inverses.