Generalized Multiresolution Analysis : Construction and Measure Theoretic Characterization

Title of dissertation: GENERALIZED MULTIRESOLUTION ANALYSIS: CONSTRUCTION AND MEASURE THEORETIC CHARACTERIZATION Juan R. Romero, Doctor of Philosophy, 2005 Dissertation directed by: Professor John J. Benedetto Department of Mathematics 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. Generalized Multiresolution Analysis: Construction and Measure Theoretic Characterization by Juan Ramon Romero Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2005 Advisory Committee: Professor John J. Benedetto, Chair/Advisor Professor Carlos A. Berenstein Professor Rebecca A. Herb Professor Raymond L. Johnson Professor Dianne P. O’Leary c © Copyright by Juan Ramon Romero 2005 In memory of Juan Ramon Romero Sr. and Elena Ioana Cioranescu.