正交指数、平移和玻尔完备化
Orthogonal exponentials, translations, and Bohr completions
摘要：
本文研究Hilbert空间$L^2(μ)$中的调和分析，其中$μ$是$\br^n$上的概率测度.统一的问题是正交（复）指数族$e_λ(x) = \exp(2πi λx)$在$L^2(μ)$的存在。这个问题反过来是连接到存在一个自然嵌入的$L^2(μ)$到$L^2$ -空间的玻尔殆周期函数的$\br^n$。特别是，我们探讨当$L^2(μ)$包含$e_λ$函数的正交基，在$\br^n$的一个合适的离散子集$λ$，即，当措施$μ$是频谱。利用局部平移群的存在性给出了有限谱集的一个新的刻画。我们还考虑措施$μ$出现的不动点（在哈钦森的意义上）的迭代函数系统（IFS），我们专门的情况下，在IFS中的函数系统由仿射和压缩映射$\br^n$。在这种情况下，我们表明，如果$μ$，然后假定频谱，那么它的分区引起的IFS在手有零重叠测量$μ$。这就解决了部分的Dallaba-Wang猜想。作为新的非重叠结果的应用，我们解决了Bernoulli卷积的谱对问题，从而提出了一个Ka-Sing Lau定理。此外，我们提出了一个新的观点，通过玻尔紧化谱措施和正交傅立叶指数。
英文摘要：
We are concerned with an harmonic analysis in Hilbert spaces $L^2(μ)$, where $μ$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_λ(x) = \exp(2πi λx)$ in $L^2(μ)$. This question in turn is connected to the existence of a natural embedding of $L^2(μ)$ into an $L^2$-space of Bohr almost periodic functions on $\br^n$. In particular we explore when $L^2(μ)$ contains an orthogonal basis of $e_λ$ functions, for $λ$ in a suitable discrete subset in $\br^n$; 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 $\br^n$. 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.
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