Uniformity of measures with Fourier frames
By Dorin Ervin Dutkay, Chun-Kit Lai. 2012
119 citations recorded by the source corpus.
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 d 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 ) .
Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e27de099f426933ed08c8fb0908950037adf0c24
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In Applied Mathematics
In Computer Vision and Pattern Recognition
In Mathematical Physics
Citations in the corpus
- Spectral measures with arbitrary Hausdorff dimensions (2014, background): “Since then, various singular spectral measures on self-similar/self-affine fractal sets have been found, see for instance [3–6,9,11,15–17,19,21,22,24,26,27,30,34].”
- Spectral property of Cantor measures with consecutive digits (2012, background): “= Xq−1 i=0 1 q µ(f−1 j (E)) (1.1) for all Borel sets E. Note that we only need to consider equal weight since nonequally weighted self-similar measures here cannot have any spectrum by Theorem 1.5 in [DL]. It is also clear that if q = 2, µ becomes the standard Cantor measure of b−1 contraction. For ”