Uniformity of measures with Fourier frames
Abbreviated Journal Title
Affine iterated function systems; Frame measures; Gabor orthonormal; bases; Hausdorff measures; Spectral measures; Tight frames; SELF-SIMILAR SETS; FRACTAL MEASURES; CANTOR MEASURES; ASYMPTOTICS; SEPARATION; DIMENSION; SYSTEMS; SERIES; TILES; Mathematics
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 gdx with Fourier frames obtained in . 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 Laba-Wang conjecture  is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang  about the structure of non-uniform Gabor orthonormal bases of the form G(g, A ,J).(C) 2013 Elsevier Inc. All rights reserved.
Advances in Mathematics
"Uniformity of measures with Fourier frames" (2014). Faculty Bibliography 2010s. 5287.