Mathematical Sciences: One Higher Dimensional Wavelets fromFractal Interpolation Functions: Construction and Applications

小行星9401352 这项补助金支持的工作重点是小波在信号和图像处理以及数值分析中的应用。 在这些领域的问题的小波方法遵循两个基本路径：开发小波从尺度方程的解决方案或通过样条。 在这项工作中要追求的第三个程序来自最近建设的分形插值函数。 该构造具有两种方法的优点，即显式的正则性结果、容易计算的内积和线性相位（如样条情形）和正交性、紧支撑和连续性（如伸缩方程情形）。 该方法也有一个多维的推广，使小波不是张量积的单变量函数。 本计画将继续研究在一维或更高维度使用分形内插函数的小波的建构与应用。 在一维中，重点将放在构造更光滑（至少可微）的紧支撑和正交小波。 在高维空间中，目标是构造连续的、紧支撑的正交小波。应用程序将利用小波的独特性质（如对称性和规律性）的图像处理。 小波表示的研究和应用是 调和分析的一个相对较新的分支，其中使用单个函数来生成希尔伯特（或其他）空间基，函数可以用这些空间基来表示。 虽然这本身并不太困难，但人们寻求具有其他属性的小波，例如平滑性，紧支持，时间和频率的局部化。 与所有这些需求相结合，人们还要求计算简单。 值得注意的是，有这样的功能。 本研究的目标是扩展和完善在这一领域的基础知识，结合现有的技术与那些平行发展的分形插值领域内。

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.