Mathematical Sciences: Wavelets and Applications

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Wavelet analysis is based on the existence of single functions whose dilations and translates form bases for function spaces used in various areas of mathematics. The theory as developed over the past decade has proved its worth through important applications to problems in signal processing, data compression, and seismic exploration, to name a few. The advantages of wavelet decompositions of functions over more traditional harmonic analysis techniques lies in the ability of properly chosen wavelets to remain local in time and frequency: a function and its Fourier transform have representations whose coefficients depend only on the values of the function in small neighborhoods distributed throughout space. Most wavelet analysis focuses on the representation or approximation of functions defined on the entire line or throughout space. There is a need to develop wavelet analyses for functions restricted to intervals. That is one of the primary goals of this project. One cannot take a wavelet theory and simply restrict it to an interval. The end points create obstacles which preclude the use of a single wavelet. At issue then is how efficiently can a wavelet theory be built on intervals in the sense of using the minimal number of auxiliary functions and maintaining the same qualities of the wavelet theory. One drawback of wavelets is their lack of symmetry. This shortcoming can be overcome by using a construction of dual Riesz bases of symmetric wavelets (one forfeits the use of a single wavelet to do all the work and replaces it by two). The adaptation of wavelets to finite intervals also opens the possibility for the construction of dual bases in this context. Very little work has been done in this direction to date, although applications to image processing appear to be very promising. One simple approach to this dual basis construction is to consider truncations of existing infinitely supported wavelets used in various applications and truncating them. These may provide interesting dual bases but they will only be of value if their conditioning numbers can be held close to unity. Work will be done investigating various examples.