9401352 Geronimo Work supported by this grant focuses on the application of wavelets to signal and image processing as well as to numerical analysis. The wavelet approach to problems in these areas has followed two basic paths: developing the wavelet from the solution of a scaling equation or via splines. A third procedure to be pursued in this work derives from recent construction of fractal interpolation functions. The construction has advantages of both of the previous methods, namely explicit regularity results, easily calculated inner products and linear phase as in the spline case and orthogonality, compact support and continuity as in the dilation equation case. The method also has a multidimensional generalization that gives wavelets which are not tensor products of univariate functions. This project continues investigations into the construction and application of wavelets using fractal interpolation functions in one and higher dimensions. In one dimension, the focus will be on constructing smoother (at least differentiable) compactly supported and orthogonal wavelets. In higher dimensions, the goal is to construct continuous, compactly supported orthogonal wavelets. Applications will be made to image processing by exploiting the unique properties (e.g. symmetry and regularity) of the wavelets. The study and applications of wavelets represents are relatively new branch of harmonic analysis in which single functions are used to generate Hilbert (or other) space bases by which functions may be represented. While this in itself is not too difficult, one seeks wavelets with additional properties such as smoothness, compact support, localization in time and frequency. Coupled with all of these demands, one also asks for computational simplicity. Remarkably, there are such functions. The goals of this research are to extend and refine basic knowledge in this field by combining existing techniques with those developed in parallel within t he field of fractal interpolation.
