An Fourier coefficient decay estimate.

f is a matrix value analytic function on $\mathbb T$, we know $latex h(f)>\alpha$, this is just mean $\forall k\in \mathbb Z$ , assume $latex | \hat f(k)|\leq e^{-|k|\alpha}$ , $g=log(f)$ for which we assume $latex g$ is a lifting of f use the inverse of ramification map ,

$M_{n\times n} \to M_{n\times n} , A \to e^A$.

Then exists $\beta=c(\alpha)>0$ such that ,

$\forall k \in \mathbb Z$, $ |\hat g(k)| \leq e^{-k\beta}$.