Orthogonal exponentials, translations, and Bohr completions
Orthogonal exponentials, translations, and Bohr completions
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DOI：
10.1016/j.jfa.2009.05.014
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发表时间：
2009-01
影响因子：
1.7
通讯作者：
D. Dutkay;P. Jorgensen;D. Han
中科院分区：
文献类型：
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作者：
D. Dutkay;P. Jorgensen;D. Han
我们关注Hilbert空间L2（μ）中的调和分析，其中μ是Rn上的一个概率测度。统一的问题是在L2（μ）中存在正交（复）指数族λ(x)=exp（2πiλx）。这个问题反过来又与L2（μ）在Rn上的玻尔概周期函数的L2空间中自然嵌入的存在性有关。特别地，我们探讨当L2（μ）包含eλ函数的正交基时，对于λ在Rn中合适的离散子集中；即，当测量μ是谱时。利用局部平移群的存在性，给出了有限谱集的一个新的表征。我们还考虑了作为迭代函数系统（IFS）不动点（在Hutchinson意义上）出现的测度μ，并且我们专门研究了IFS中的函数系统由Rn中的仿射映射和收缩映射组成的情况。在这种情况下，我们证明，如果μ被假设为谱，那么由手头的IFS引起的分区在μ测量中具有零重叠。这解决了Łaba-Wang猜想的部分问题。作为新的非重叠结果的一个应用，我们用这种方法推进了刘家声的一个定理，解决了伯努利卷积的谱对问题。此外，我们通过玻尔紧化提出了光谱测度和正交傅立叶指数的新观点。
We are concerned with an harmonic analysis in Hilbert spaces L2(μ), where μ is a probability measure on Rn. The unifying question is the presence of families of orthogonal (complex) exponentials eλ(x)=exp(2πiλx) in L2(μ). This question in turn is connected to the existence of a natural embedding of L2(μ) into an L2-space of Bohr almost periodic functions on Rn. In particular we explore when L2(μ) contains an orthogonal basis of eλfunctions, for λ in a suitable discrete subset in Rn; i.e, when the measure μ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures μ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in Rn. We show in this case that if μ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in μ. This solves part of the Łaba–Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.