Wavelets: Theory and Applications
The aim of this thesis is to present an overview of the Fourier transform, including the continuous Fourier transform, the l2(Z) transform (Fourier series), the discrete Fourier transform, and the windowed Fourier transform. The shortcomings of these has led to the recently discovered wavelet transform, which has corresponding continuous and discrete versions. I present the theory behind these wavelet transforms, and present some of their relevant applications. Multiresolution Analysis (MRA) will be highlighted as a tool of great potential utility in other fields such as Physics, Biology, Geology, Computer Science, Medicine, and Engineering. I conclude this thesis with a discussion of three "signature" recognition applications of the discrete wavelet transform MRA to gamma-ray spectrum analysis, amphibian identification, and magnetic dipole detection.