Mathematical Sciences: Wavelets and Applications

小波分析基于单个函数的存在，这些函数的伸缩和平移是用于数学各个领域的函数空间的形式基。在过去十年中发展起来的理论已经通过在信号处理、数据压缩和地震勘探等问题上的重要应用证明了它的价值。与更传统的调和分析技术相比，函数的小波分解的优势在于，适当选择的小波能够在时间和频率上保持局部：函数及其傅立叶变换具有其系数仅取决于分布在整个空间的小邻域中的函数值的表示。大多数小波分析侧重于定义在整个直线或整个空间上的函数的表示或近似。需要为限制在区间内的函数开发小波分析。这是该项目的主要目标之一。人们不能把小波理论简单地限制在一个区间内。端点会造成障碍，从而无法使用单个子波。因此，在使用最少数量的辅助函数并保持小波理论的相同性质的意义上，如何有效地在区间上构建小波理论是问题所在。小波的一个缺点是它们缺乏对称性。这一缺点可以通过使用对称小波的对偶Riesz基的构造来克服(一个人放弃了使用一个小波来做所有的工作，而用两个小波来代替它)。小波对有限区间的适应也为在此背景下构造对偶基打开了可能性。到目前为止，在这个方向上做的工作很少，尽管在图像处理方面的应用似乎非常有前途。这种对偶基构造的一种简单方法是考虑各种应用中使用的现有无限支撑子波的截断并截断它们。这些可能提供了有趣的双基，但只有当它们的条件数能够保持接近统一时，它们才有价值。将对各种例子进行调查。

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.