Mathematical Sciences: Wavelets Based on Several Scaling Functions and Related Applications

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9503282 Wang This project will involve research in the theory and applications of wavelets. The problems to be considered include: (1) the construction and characterization of wavelets of Battle-Federbush type and their associated wavelet packets; (2) the general construction of multiwavelets from an orthogonal multiresolution analysis generated by scaling vectors; (3) determining properties of certain classes of solutions of matrix refinement equations; (4) applications of fractal multiwavelets to second order hyperbolic partial differential equations; and (5) adaptations of fast wavelet transformations by incorporating multiwavelets in order to study nonlinear integral equations. This project is being supported under the Research in Undergraduate Institutions (RUI) activity. The RUI activity is part of the Foundation's effort to help assure a broad base for science and engineering research, and thereby enhance the scientific and technical training of students in undergraduate institutions. The specific objectives of the RUI program are to: (1) support high quality research by faculty with active involvement of undergraduate students, (2) strengthen the research environment in academic departments that are oriented primarily toward undergraduate instruction, and (3) promote the integration of research and education at predominantly undergraduate institutions. This project will involve research on the theory of multiwavelets. Multiwavelet theory involves replacing the classical dilation equation of wavelet theory by a matrix dilation equation. The relationship of the dilation equation and the various features of the wavelet bases generated will be investigated. The use of wavelets for the numerical solution of partial differential and integral equations will also be studied; these techniques will be compared to more classical algorithms.