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

TL;DR

This work develops a method to approximate increasing smooth functions by the argument of meromorphic inner functions under a local doubling condition, yielding controlled phase derivatives up to polynomial loss. The authors then leverage this construction to (i) derive a Beurling-type density criterion for zero sets of Toeplitz kernels with unimodular real-analytic symbols and (ii) produce admissible Beurling-Malliavin majorants in model spaces generated by one-component inner functions. The framework unifies Toeplitz-kernel theory and model-space majorants under regular locally doubling weights, broadening applicability to spectral problems and uncertainty principles in one dimension. The results extend prior work by offering general, quantitative control over phase, zeros, and majorants in a broad class of inner- and model-spaces contexts.

Abstract

Let $f$ be a strictly increasing smooth function, such that $f'$ is comparable to a weight $α'$ 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 $α'$ up to polynomial loss. We give two applications of this result. The first is a sufficient density condition for a set $Λ$ 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.

Theorems & Definitions (25)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof