# Wavelet and gabor frames and bases approximation, sampling and applications

Abstract: This thesis is devoted to both theoretical and practical aspects of applied mathematics. It consists of three main parts: Part I consists of an application-oriented introduction to the theory of frames and bases for separable Hilbert spaces, as well as an introduction to the main tools used in the remaining Chapters: Time-frequency analysis, Gabor frames and wavelet frames. Part II contain five publications in the fields of approximation theory, sampling and perturbation stability. One paper and one research report consider different estimates of the error (measured in L^p, Besov or Triebel-Lizorkin norm) when a function is projected on certain so-called shift-invariant spaces. This is closely connected to a certain class of wavelet subspace sampling problems (containing the classical Shannon sampling theorem as a special case), prefiltering of the discrete wavelet transform and a certain perturbation stability theorem called the Kadec 1/4-theorem. These are the topics of the three remaining papers in Part II. Chapter 3 contains estimates of the L^p-norm error in certain projections of a function f onto a shift-invariant space V_j, spanned by translated copies of some given function. In Chapter 4, we propose a method which is well-suited for studying irregular sampling problems in such spaces. The main advantage of our method is that it provides an intuitive understanding and relatively simple proofs of some studied problems. The method builds on a study of an equivalent coefficient mapping, which we also use in Chapter 5. There we propose an improved low-complexity approximation of an often neglected prefilter which is needed when the discrete wavelet transform is used for analysis of real-world (i.e., non-discrete) signals. Chapter 6 consists of further results and remarks. First we prove some new results related to Chapter 3, but now the error estimates are measured in other function space norms. Then we conclude Part II with a collection of remarks on some recent proofs of some perturbation theory theorems, including the Kadec 1/4-theorem, which is equivalent to the classical Shannon sampling problem and a special case of the problems studied in Chapter 4. Part III contains one selected publication from each the two practical applications which I have been working with during my years as Ph.D. student: VDSL signal transmission and bearing condition monitoring. The first paper is one of four papers which were included and defended in my licentiate thesis. They describe the solutions to some important problems during the development of a duplex method (called Zipper) for very high speed digital communication in ordinary unshielded telephone copper wires. This was done together with Telia Research and the Division of Signal processing and the end-product (VDSL modems for up to 52 Mbit/s) is currently under development by the French-Italian company ST Microelectronics (former SGS-Thomson). The paper included here describes a patented method for reducing the interference that the unshielded copper wires experience from high-power narrowband radio transmissions, such as radio amateurs. The last paper, finally, is an overview article about bearing condition monitoring. Here the main problem is to find a method for predicting bearing failures by analysis of vibration measurements from rotating machines. The paper contains the main results of a co-operation with (among others) Nåiden Teknik and three forestry combines. We argue that time-frequency analysis based methods are well suited for this task and compare such methods with different old and new methods, using a large number of (mainly industrial environment) test signals.

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