Mathematical Sciences: Wavelets and Applications

小波分析是基于单 函数，其伸缩和平移形成函数的基 空间在数学的各个领域中使用。 理论作为 在过去十年中发展起来的技术已经证明了其价值， 信号处理、数据处理、 压缩和地震勘探，仅举几例。 的 函数的小波分解的优点 传统的谐波分析技术在于 适当选择小波，以保持时间和频率的局部性：a 函数及其傅里叶变换具有表示， 系数仅取决于函数的值， 分布在整个空间的街区。 最小波 分析的重点是表示或近似 定义在整条线或整个空间上的功能。 那里 是一个需要发展小波分析的功能限制， 的间隔 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 interval. 端点会产生阻碍使用的障碍 一个单一的wavelet 接下来的问题是， 小波理论建立在区间上， 最小数量的辅助功能并保持相同 小波理论的性质。 小波的一个缺点是缺乏对称性。 这 利用对偶Riesz构造可以克服上述缺点 对称小波基（一个放弃使用一个单一的 小波做所有的工作，并将其替换为2）。 的 小波对有限区间的适应也打开了 在这种情况下建立双重基地的可能性。 迄今为止，在这方面所做的工作很少， 尽管图像处理的应用看起来非常 很有希望 一个简单的方法来构造这个对偶基 是考虑截断现有的无限支持 在各种应用中使用的小波，并截断它们。 这些 可能会提供有趣的双基地，但他们只会有价值 如果它们的条件数能保持接近1的话 工作 我们将研究各种各样的例子。

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.