Fourier标架测度的一致性
Uniformity of measures with Fourier frames
摘要：
我们研究傅立叶框架，更一般地说，不同的概率测度的框架措施。我们证明了，如果一个措施有一个相关的框架措施，那么它必须有一定的一致性，在这个意义上的重量是相当均匀地分布在其支持。更精确地说，通过考虑测度及其平移的某些绝对连续性质，我们用\cite{Lai11}中得到的Fourier标架恢复了绝对连续测度$g\, dx$的特征.此外，我们证明了框架的界限被推到了本质的下确界和上确界的功能$g$。这也表明，绝对连续谱测度支持一个集$Ω$，如果他们存在，必须是标准的勒贝格措施$Ω$上的乘法常数。然后，我们研究仿射迭代函数系统（IFS），我们证明了，如果一个IFS没有重叠承认一个框架测度，那么概率权重都是相等的。此外，我们还证明了当自相似测度是绝对连续的时，Dahaba-Wang猜想\cite{MR1929508}成立.最后，我们将提出一个新的方法来解决刘和王\cite{LW}猜想的非均匀Gabor规范正交基的结构的形式${\mathcal G}(g,Λ,{\mathcal J})$。
英文摘要：
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 $g\, dx$ with Fourier frames obtained in \cite{Lai11}. 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 \cite{MR1929508} is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang \cite{LW} about the structure of non-uniform Gabor orthonormal bases of the form ${\mathcal G}(g,Λ,{\mathcal J})$.
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