Wavelet Transforms and Localization Operators

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This book is based on lectures from the Global Analysis Research Center at Seoul National University and Peking University in 1999 and 2000. Preliminary versions have been used for graduate courses at York University. It focuses on wavelet transforms and localization operators within the framework of infinite-dimensional and square-integrable representations of locally compact and Hausdorff groups. The wavelet transforms examined include those from the Weyl-Heisenberg group and the affine group, serving as foundational elements for localization operators. A central theme is the spectral theory of these transforms and operators, particularly through Schatten-von Neumann norm inequalities. Several chapters discuss product formulas for specific localization operators, such as Daubechies operators and wavelet multipliers. This work follows the author’s previous books on pseudo-differential operators and Weyl transforms, highlighting that localization operators on the Weyl-Heisenberg group are indeed Weyl transforms, which are pseudo-differential operators. The first chapter outlines the book's perspective and organization. It is intended for readers with a background in measure theory and functional analysis, along with some knowledge of group theory and general topology at the undergraduate level.