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 ^ -norm of the sequence of the Fourier coefficients of B^n 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.