On Fourier frame of absolutely continuous measures

TL;DR

This paper characterizes when absolutely continuous measures on R^n admit Fourier frames, linking it to bounds on their Radon-Nikodym derivatives, and applies this to self-similar measures and Bernoulli convolutions.

Contribution

It provides a necessary and sufficient condition for Fourier frames of absolutely continuous measures and characterizes self-similar measures that admit Fourier frames.

Findings

Fourier frames exist iff the Radon-Nikodym derivative is bounded above and below.

Self-similar measures admitting Fourier frames are characteristic functions of self-similar tiles.

Bernoulli convolutions with parameters 1/2<λ<1 do not admit Fourier frames.

Abstract

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support. As a consequence, we prove that if an equal weight absolutely continuous self-similar measure on ${\Bbb R}^1$ admits Fourier frame, then the measure must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere $1/2<\lambda<1$, the $\lambda$-Bernoulli convolutions cannot admit Fourier frames.