Wavelet and Fourier bases on Fractals

In this thesis we first develop a geometric framework for spectral pairs and for orthonormal families of complex exponential functions in L2-spaces with respect to a given Borel probability measure that is compactly supported. Secondly, we develop wavelet bases on L2-spaces based on limit sets of different iteration systems. In the framework of spectral pairs we consider families of exponential functions with a countable index set G which difference set is equal to all integers, and we determine the L2-spaces in which these functions are orthonormal or constitute a basis. We also consider invariant measures on Cantor sets and study for which measures we have a family of exponential functions that is an orthonormal basis for the L2-space with respect to this measure. For the case of Cantor sets the families of exponential functions are obtained via Hadamard matrices. For the study of wavelet bases, we set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the translation by integers of a non-atomic measure supported on the limit set of such a system and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets and then apply these abstract results to Markov Interval Maps. Even though, in our setting, the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets. We also set up a multiresolution analysis for enlarged fractals in one and two dimensions, which are sets arising from fractals that are generated by iterated function systems, so that the enlarged fractals are dense in the lin or plane, respectively. The measure supported on the fractal is also extended to a measure on the enlarged fractal. We then construct a wavelet basis via multiresolution analysis on this L2-space with respect to the measure having the enlarged fractal as the support, with the characteristic function of the original fractal as the father wavelet which gives us via the multiresolution analysis the wavelet basis for the L2-space. In this construction we have two unitary operators. Finally, we also apply the wavelet bases on enlarged fractals in two dimensions to image compression

Multiresolution approximation of the vector fields on T^3

Multiresolution approximation (MRA) of the vector fields on T^3 is studied. We introduced in the Fourier space a triad of vector fields called helical vectors which derived from the spherical coordinate system basis. Utilizing the helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3 and the Beltrami decomposition that decompose the space of solenoidal vector fields into the eigenspaces of curl operator. In the course of proof, a general construction procedure of the divergence-free orthonormal complete basis from the basis of scalar function space is presented. Applying this procedure to MRA of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity and regularity of vector wavelets. It is conjectured that the solenoidal wavelet basis must break r-regular condition, i.e. some wavelet functions cannot be rapidly decreasing function because of the inevitable singularities of helical vectors. The localization property and spatial structure of solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's wavelet) are also investigated numerically.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy

Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L^2(R^d) which have generalized multiresolutions. This means that the initial resolution subspace V_0 in L^2(R^d) is not singly generated. As a result, the representation of the integer lattice Z^d restricted to V_0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R^d can be constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos in Sections 1 and 4, v3 adds a number of references on GMRA theory and wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and two more reference

Classification of Generalized Multiresolution Analyses

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions $m$ and matrix-valued filter functions $H$. Given a natural number valued function $m$ and a system of functions encoded in a matrix $H$ satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function $m$ and filter system $H$. An equivalence relation on GMRAs is defined and described in terms of their associated pairs $(m,H)$. This classification system is applied to classical examples in $L^2 (\mathbb R^d)$ as well as to previously studied abstract examples.Comment: 18 pages including bibliograp