圆和实直线上的小波变换：统一的群论处理
Wavelet Transform on the Circle and the Real Line: A Unified Group-Theoretical Treatment
摘要：
我们提出了一个统一的群论推导的连续小波变换（CWT）的圆$\mathbb S^1$和实数线$\mathbb{R}$，以下的一般形式主义的相干态（CS）相关的酉平方可积（模子群，可能）表示的组$SL(2,\mathbb{R})$。描述了在信号$L^2(X,dx)$的空间上获得仿射变换的群$G$的酉表示的一般过程，将载波空间$X$与（一阶或高阶）“极化子代数”${\cal P}_X$相关联。给出了$\mathbb S^1$上小波的容许性和连续框架条件，并讨论了群压缩下的欧氏极限。
英文摘要：
We present a unified group-theoretical derivation of the Continuous Wavelet Transform (CWT) on the circle $\mathbb S^1$ and the real line $\mathbb{R}$, following the general formalism of Coherent States (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group $SL(2,\mathbb{R})$. A general procedure for obtaining unitary representations of a group $G$ of affine transformations on a space of signals $L^2(X,dx)$ is described, relating carrier spaces $X$ to (first or higher-order) ``polarization subalgebras'' ${\cal P}_X$. We also provide explicit admissibility and continuous frame conditions for wavelets on $\mathbb S^1$ and discuss the Euclidean limit in terms of group contraction.
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