Title

Uniformity Of Measures With Fourier Frames

Keywords

Affine iterated function systems; Frame measures; Gabor orthonormal bases; Hausdorff measures; Spectral measures; Tight frames

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 gd x with Fourier frames obtained in [24]. 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 Ω, if they exist, must be the standard Lebesgue measure on Ω 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 Łaba-Wang conjecture [20] is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang [29] about the structure of non-uniform Gabor orthonormal bases of the form G(g,Λ,J). © 2013 Elsevier Inc.

Publication Date

2-15-2014

Publication Title

Advances in Mathematics

Volume

252

Number of Pages

684-707

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.aim.2013.11.012

Socpus ID

84890348167 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84890348167

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