On Fourier frame of absolutely continuous measures |
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Authors: | Chun-Kit Lai |
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Institution: | Department of Mathematics, The Chinese University of Hong Kong, Hong Kong |
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Abstract: | Let μ be a compactly supported absolutely continuous probability measure on Rn, we show that L2(K,dμ) admits a Fourier frame if and only if its Radon-Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R1 and L2(K,dμ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<1, the L2 space of the λ-Bernoulli convolutions cannot admit a Fourier frame. |
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Keywords: | Absolute continuity Bernoulli convolution Beurling density Fourier frame Self-similar measure |
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