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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Continuous and discrete Fourier frames for fractal measures


Authors: Dorin Ervin Dutkay, Deguang Han and Eric Weber
Journal: Trans. Amer. Math. Soc. 366 (2014), 1213-1235
MSC (2010): Primary 28A80, 28A78, 42B05
Published electronically: August 2, 2013
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Abstract: Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $ \mathbb{R}^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure, then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular, we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure, then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.


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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
Email: Dorin.Dutkay@ucf.edu

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
Email: deguang.han@ucf.edu

Eric Weber
Affiliation: Department of Mathematics, 396 Carver Hall, Iowa State University, Ames, Iowa 50011
Email: esweber@iastate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05843-6
PII: S 0002-9947(2013)05843-6
Keywords: Plancherel theorem, frame, Bessel, Fourier series, Hilbert space, fractal, self-similar, iterated function system
Received by editor(s): November 9, 2011
Published electronically: August 2, 2013
Additional Notes: This research was supported in part by the National Science Foundation grant 1106934
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.