Coherent States, Wavelets and Their Generalizations

S. Twareque Ali

This book presents a survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. The point of view is that both the theories of both wavelets and coherent states can be subsumed into a single analytic structure. Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent otherwise obscure properties of wavelets and of coherent states. Many concrete examples, such as semisimple Lie...