Generalized Multiresolution Analysis: Construction and Measure Theoretic Characterization

Abstract

In this dissertation, we first study the theory of frame multiresolution analysis (FMRA) and extend some of the most significant results to d - dimensional Euclidean spaces. A main feature of this theory is the fact that it was successfully applied to narrow band signals; however, the theory does have its limitations. Some orthonormal wavelets may not be obtained by the methods of FMRA. This is because non-MRA orthonormal wavelets have nonconstant dimension functions. This means that the number of scaling functions needed is more than one. The appropiate tools for non-MRA wavelets are the generalized multiresolution analyses (GFMRA, GMRA) theories developed by Manos Papadakis and Lawrence Baggett. At the end, we unify both theories by finding an explicit formula for an important map. Our approach also permits us to give a short and elegant proof of a classical result about a special type of decomposition in shift-invariant space theory.