Inner approximations of doubling weights with applications to Beurling-Malliavin theory in Toeplitz kernels

Abstract: Let $f$ be a strictly increasing smooth function, such that $f'$ is comparable to a weight $\alpha'$ which is locally doubling and satisfies a non-triviality condition to be explained in the paper. We construct a meromorphic inner function $J$, such that $f-\arg(J)$ is bounded and $\arg(J)'$ is comparable to $\alpha'$ up to polynomial loss. We give two applications of this result. The first is a sufficient density condition for a set $\Lambda$ to be a zero set for a Toeplitz kernel with real analytic and unimodular symbol. Our second application is to describe a class of admissible Beurling-Malliavin majorants in model spaces. The generality considered here lets us treat most cases of model spaces generated by one-component inner functions.