小波分解与Martin边界
Decomposition of wavelet representations and Martin boundaries
摘要：
我们研究了一类酉表示与小波分析，小波表示的分解问题，但我们的框架是更广泛的，并已应用到多尺度扩展动力系统理论中产生的不可逆自同态。我们的主要结果提供了一个直接的积分分解的一般小波表示，我们解决了朱迪思包装所提出的问题。这需要一个直接的整体分解的一般小波表示。我们进一步给出了一个详细的分析的措施，有助于分解成不可约表示。我们证明结果相关的马丁边界，相关的小波滤波器和诱导随机行走的理解，以及类的调和函数。我们的设置需要表示建立从某些有限到一的自同态$r$在紧度量空间$X$，我们研究了它们的膨胀诱导的自同构。我们的小波表示是由扩张自同构形成的协变系统。它们依赖于分配的措施$μ$对$X$。已知当数据$(X, r, μ)$被给定时，相关的小波表示通常是可约的。通过引入与$(X, r)$相关的小波滤波器，我们在$X$中建立随机游动、路径空间测度、调和函数和相关的Martin边界。我们构造的螺线管$(X_\infty, r_\infty)$，从$(X, r)$构建的措施。证明了$r_\infty$在Hilbert空间$\H$和代数$L^\infty(X)$的表示$π$上诱导酉算子$U$，使得对$(U, r_\infty)$与相应的表示$π$一起形成$C^*$ -代数意义下的叉积.我们注意到，传统的小波表示属于这个更广泛的框架内的$(\H, U, π)$协变交叉产品。
英文摘要：
We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random-walks, as well as classes of harmonic functions. Our setting entails representations built from certain finite-to-one endomorphisms $r$ in compact metric spaces $X$, and we study their dilations to automorphisms in induced solenoids. Our wavelet representations are covariant systems formed from the dilated automorphisms. They depend on assigned measures $μ$ on $X$. It is known that when the data $(X, r, μ)$ are given the associated wavelet representation is typically reducible. By introducing wavelet filters associated to $(X, r)$ we build random walks in $X$, path-space measures, harmonic functions, and an associated Martin boundary. We construct measures on the solenoid $(X_\infty, r_\infty)$, built from $(X, r)$. We show that $r_\infty$ induces unitary operators $U$ on Hilbert space $\H$ and representations $π$ of the algebra $L^\infty(X)$ such that the pair $(U, r_\infty)$, together with the corresponding representation $π$ forms a cross-product in the sense of $C^*$-algebras. We note that the traditional wavelet representations fall within this wider framework of $(\H, U, π)$ covariant crossed products.
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