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arXiv · 1103.4691

On Fourier frame of absolutely continuous measures

Chun-kit Lai
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Abstract

Let $μ$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $μ$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support. As a consequence, we prove that if an equal weight absolutely continuous self-similar measure on ${\Bbb R}^1$ admits Fourier frame, then the measure must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere $1/2<λ<1$, the $λ$-Bernoulli convolutions cannot admit Fourier frames.

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Chun-kit Lai. 2011-03-24. On Fourier frame of absolutely continuous measures. https://arxiv.org/abs/1103.4691

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