On Fourier frame of absolutely continuous measures

By Chun-Kit Lai. 2011

61 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42e2f9045373db4f4641678951eec21775edde68

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In Applied Mathematics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Qiyu Sun
5. Ka-Sing Lau
6. Dorin Ervin Dutkay
7. Li-Xiang An
8. Xiaoye Fu

In Mathematical Physics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Qiyu Sun
5. Ka-Sing Lau
6. Dorin Ervin Dutkay
7. Li-Xiang An
8. Xiaoye Fu

Citations in the corpus

• Exponential spectra in L2(μ) (2013, background): “In another general situation, Lai [13] proved a sharp result that if μ is absolutely continuous with respect to the Lebesgue measure, then it is an F-spectral measure if and only if its density function is essentially bounded above and below on the support.”
• Spectral property of Cantor measures with consecutive digits (2012, background): “able amount of literatures studying spectral measures and other generalized types of Fourier expansions like the Fourier frames and Riesz bases ([DHJ], [DHSW], [DL], [HLL], [IP], [JKS] [L a], [L aW], [Lai], [Li1], [Li2], and the references therein). In [HuL], Hu and Lau made a start in studying the ”
• 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].”
• Uniformity of measures with Fourier frames (2012, methodology): “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 dx with Fourier frames obtained in [Lai11].”