## Retrospective Theses and Dissertations

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# Pointwise Convergence for Wavelet Expansions

## Author's School

Graduate School of Arts & Sciences

## Author's Department/Program

Mathematics

Richard H. Rochberg, Mitchell H. Taibleson, Carl M. Bender, Daniel R. Fuhrmann, Heinz M. Schaettler, Guido L. Weiss, M. Victor Wickerhauser

English (en)

Spring 5-15-1992

## Degree Type

Restricted Access Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Abstract

A wavelet basis is an orthonormal basis of smooth functions generated by dilations by 2-m and translations by n2-m of a single function. While the discovery of such bases has only occurred in the last ten years or so, earlier results in Harmonic Analysis have provided the groundwork for this new topic. Wavelet theory is a culmination of many ideas in this area. A brief history of some of the ideas for wavelets will be given in Chapter 1. Wavelet theory has caught the interest of many mathematicians as well as engineers and other scientists who are interested in its many applications. Applications for wavelets include uses in medicine for faster and sharper scanners for medical diagnosis, more efficient computer storage techniques for large amounts of data, and sharper audio and visual signals with less information needed. Wavelets are beneficial in these applications because of the highly efficient ways in which a wavelet series can represent a signal. This allows for better reproductions of functions with less data needed. More details of this will be given in the chapters to follow. In this paper, we will look at pointwise convergence properties of various types of wavelets. Because the wavelet bases are made up of dilations of a function, the wavelet functions are able to zoom in on any part of a function being evaluated. In Chapter 1, background for a motivation for wavelets will be given along with examples of various types of wavelet bases which will be explored in more detail in following chapters. The underlying structure, which has been hinted at in this Introduction, will be discussed in greater detail. Also, a comparison between a local property of wavelet expansions compared to expansions by the standard and a windowed Fourier transform will be given. In Chapter 2, pointwise convergence properties will be found for wavelets in S(R ), that is wavelets with rapid decrease. The convergence for the wavelets in S(R) of P.G. Lemarie and Y. Meyer [Le-Me] will specifically be examined. In particular, convergence of a wavelet expansion at a Lebesgue point will be proved. Rates of convergence for wavelet expansion of a function at points with specific smoothness conditions will also be found. In Chapter 3, similar results to those obtained in Chapter 2 will be found for wavelets with exponential decay. In particular spline wavelets on L2[0, 1) and on L2(R ) will be used as examples. These wavelets were developed by J.O. Stromberg [St], G. Battle [Ba] and P.G. Lemarie [Le]. Finally, in Chapter 4 the Gibbs effect for wavelet expansions will be explored. The Gibbs effect is the phenomenon in which a partial sum expansion of a function overshoots or undershoots the original function near a jump discontinuity of the function. A condition to determine if there is a Gibbs effect for a general wavelet expansion will be given. This condition will then be used to prove that a Gibbs effect does exist for at least some compactly supported wavelets. Finally, results from a computer analysis will be given which were used to estimate the size of the Gibbs effects for expansions of functions by some compactly supported wavelets. The specific examples used in this chapter deal with the Haar system and I. Daubechies's compactly supported wavelets [Dal]. Results from the work of others will be listed as propositions. New results given in this dissertation will be given as lemmas, theorems and corollaries. Definitions in this paper are commonly known and are the work of others. A good reference for many of the definitions used in this paper is [Fr-Ja-We].