In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.

在本文中，我们介绍了一种用于小波变换的新的本地化框架，例如一维小波变换和Shearlet变换。我们的目标是设计从某种意义上促进稀疏性的非自适应窗口功能。为此，我们介绍了一个用于分析窗口函数的本地化方面的框架。我们的本地化理论在两个方面不同于传统理论。首先，我们区分组生成器和测量本地化的运算符（称为可观察值）。其次，我们定义整个信号变换的不确定性，而不是定义单个窗口的不确定性。我们表明，在信号空间中窗函数的不确定性与小波变换的再现核在相空间中的定位密切相关。结果是，