  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 based on a window 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.
