Abstract Constructions of GMRAs
This chapter presents abstract constructions of GMRAs and wavelets using direct sums and direct limits. As a first example, we take up super-wavelets, which were developed in direct sum spaces to handle the application of multiplexing, the sending of multiple signals on a carrier at the same time. Direct limits, the second construction technique we discuss, were first used to build classical MRA’s and wavelets from filters, and later generalized to construct abstract GMRA’s from multiplicity functions. Finally we present a technique that uses direct sums to find a classifying set for all GMRA’s with finite multiplicity function and Haar measure class.
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