# Wavelet Analysis

**Abstract:** A discussion of
orthonormal wavelet bases and wavelet series, utilising the notion of
multiresolution approximations. I start by discussing multiresolution
approximations and show how to find an orthonormal basis of the
generating space, consisting of translates of a `scaling function'
φ. The notion of orthornormal wavelet bases is then introduced,
and it is shown how one can go from a multiresolution approximation to
a wavelet basis with a scaling function, and vice-versa. The article
then turns to the description of wavelet bases of compact support and
multi-dimensional wavelet bases (using the tensor product
method). Finally, the regular wavelets of compact support are seen to
form an unconditional basis of the space
H^{1}(**R**^{n}) of Stein and Weiss, this being an
interesting subspace of L^{1}(**R**^{n}).

This is my MSci project, which I undertook in the fourth year
of my undergraduate degree at Imperial College, London. It is
concerned with mathematical entities known as wavelets, and uses the
wavelets to investigate a space called H^{1}(**R**^{n}).

Though the title of the project is `Elementary' wavelet analysis,
you still do need to know some maths to understand it! In particular,
you should be aquainted with undergraduate functional analysis. If
you'd prefer something less technical, why don't you try The
Engineers Ultimate Guide to Wavelet Analysis. This Internet
article was written by a non-mathematician and designed for
non-mathematicians, so anyone (well any scientist or engineer) should
find it quite accessible. If you want to read more on the topic, then
another not-so-technical resource is G. Kaiser,
*A Friendly Guide to Wavelets*, Birkhauser, Boston 1994.
A good introductory volume for the more mathematically minded amongst
you is C. K. Chui, *An Introduction to
Wavelets*, Academic Press Inc., New York, 1992. But the
book I referred to mostly whilst writing my project was Y. Meyer, *Wavelets and operators*, Cambridge
University Press, Cambridge, 1992. It's a pretty tough going
book and I wouldn't recommend it unless you're really up to scratch on
your real analysis; none the less it is required reading if you
seriously want to understand the deeper theoretical foundations of the
wavelet theory.

In case you meet the above requirement and are interested in reading further, you might first like to know a little about what wavelets are and how they can be used.

A one-dimensional wavelet basis is a basis of
L^{2}(**R**) (depending upon the author, the basis may or
may not be required to be orthonormal) and equally an n-dimensional
wavelet basis is a basis of L^{2}(**R**^{n}), in
much the same manner the complex exponentials form an orthonormal
basis of L^{2}(0,2π). So indeed wavelet analysis shares
some similarities with Fourier analysis. However if you read the
article, you'll find that wavelets really kick arse, compared with
Fourier analysis.