Uniformity of measures with Fourier frames

TL;DR

This paper investigates the uniformity conditions of measures that admit Fourier frames, showing that such measures must be uniformly distributed, and explores implications for affine IFSs, spectral measures, and Gabor bases.

Contribution

It establishes that measures with Fourier frames exhibit uniformity, proves the Lebesgue measure is essentially unique for spectral measures on a set, and confirms the equal weight condition for IFSs without overlap.

Findings

Measures with Fourier frames are uniformly distributed on their support.

Absolutely continuous spectral measures are proportional to Lebesgue measure.

IFS with no overlap and a frame measure must have equal probability weights.

Abstract

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. To be more precise, by considering certain absolute continuity properties of the measure and its translation, we recover the characterization on absolutely continuous measures $g\, dx$ with Fourier frames obtained in \cite{Lai11}. Moreover, we prove that the frame bounds are pushed away by the essential infimum and supremum of the function $g$. This also shows that absolutely continuous spectral measures supported on a set $\Omega$, if they exist, must be the standard Lebesgue measure on $\Omega$ up to a multiplicative constant. We then investigate affine iterated function systems (IFSs), we show that if an IFS with no overlap admits a frame measure then the probability weights are all equal. Moreover, we also show that the {\L}aba-Wang conjecture \cite{MR1929508} is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang \cite{LW} about the structure of non-uniform Gabor orthonormal bases of the form ${\mathcal G}(g,\Lambda,{\mathcal J})$.