Mathematical Sciences: Wavelets, Mulivariate Splines, and Radial Functions

调查人员在三个领域进行研究， 近似理论：小波分析，多元样条和 小波和径向基函数。 在小波的研究中， 除了数学分析之外，一个主要的重点是应用 实时（或在线）时频分析。 因此 研究人员研究分析和算法， 样条小波和非正交的计算问题 小波包 特别是，他们将设计一个 沿着卡尔曼滤波器的直线沿着预测校正算法 实现了样条小波分解的无截断 并且处理诸如白色噪声的噪声过程。 的 对多元样条的研究也扩展到构造 使用Box Splines的多变量小波。 一个最 困难的问题是研究最小支持， 上升到具有最小长度的重建序列。 再次 将实施预测-校正算法以避免 分解序列的截断。“径向基 函数”是一个工具，允许拟合常规和 离散数据的连续可微函数。某些 与插值和逼近相关的矩阵 与数据拟合相关的过程出现。 的 研究人员将研究这些基质的“稳定性”。 他们 还考虑到如何迅速的问题的各个方面， 近似表面接近下面的表面， 分散的数据被采样为数据点的数量 增大 小波变换是一种切割数据或 函数以及它们的频谱到不同的频率 组件，然后使用分辨率研究每个组件 匹配到合适的规模。 例如，给定一个信号， 一个感兴趣的是“切割”其频率内容， 当地时间信息。 这类似于音乐符号， 告诉演奏者在任何给定时刻演奏哪个音符。 小波、小波变换和小波包有助于 分析信号和数据。 小波分析在数据分析中很有用 减少，这已经应用于高清晰度电视。 使用小波，调查人员的目的是分解和 尽可能简单和准确地重建信号。 另一个目的是代表产生 通过“简单”径向函数的方式对分散的数据进行处理。 这 允许人们分析和推断数据，这很重要， 用于人工学习设备。

The investigators undertake research in three areas of approximation theory: wavelet analysis, multivariate splines and wavelets, and radial basis functions. In the study of wavelets, beyond the mathematical analysis a major focus is on applications to real-time (or on-line) time-frequency analysis. Hence the investigators study the analysis and algorithmic and computational aspects of spline-wavelets and non-orthogonal wavelet packets. In particular, they will design a predictor-corrector algorithm along the line of the Kalman filter to implement the spline-wavelet decomposition without truncation and to take care of noise processes such as white noise. The research on multivariate splines also extends to the construction of multivariate wavelets using box splines. One of the most difficult problems is the study of minimum supports that give rise to reconstruction sequences with minimum length. Again a predictor-corrector algorithm will be implemented to avoid truncation of the decomposition sequences. "Radial basis functions" is a tool that allows fiting both regular and scattered data by continuously differentiable functions. Certain matrices associated with interpolation and approximation processes arise in connection with the data fitting. The investigators will study the "stability" of these matrices. They also consider various aspects of the question of how rapidly the approximating surfaces approach the underlying surface from which the scattered data are sampled as the number of data points increases. The wavelet transform is a tool that cuts up data or functions as well as their spectra into different frequency components and then studies each component with a resolution matched to the appropriate scale. For example, given a signal, one is interested in "cutting" up its frequency content with local time-information. This is similar to music notation, which tells the player which notes to play at any given moment. Wavelets, wavelet transforms, and wavelet packets help in analyzing signals and data. Wavelet analysis is useful in data reduction, which already has been applied to high-definition T.V. Using wavelets, the investigators aim at decomposing and reconstructing signals as simply and accurately as possible. Another aim is to represent the underlying processes generating scattered data by means of "simple" radial functions. This allows one to analyze and extrapolate data, which is important for artificial learning devices.