Approximations of Functions from Scattered Data: Theory and Applications

这位作者特别感兴趣的是用正定函数和相关函数来表示和分析n维空间和某些紧致流形(如m维球面)上的散乱数据。目前已有一些关于紧致流形上正定(及相关)函数逼近的结果，但在n维欧氏空间上尚无类似的结果。例如，拟插值逼近方法，Marcinkiewicz-Zygmund不等式，导致与散乱点有关的求积公式，一些多水平逼近方法的逼近速度估计，以及许多已知在球面上成立的结果，在n维欧氏空间上并不具有良好的对应性。为了更好地理解正定函数的逼近势及其应用，在这些空间上建立一些相应的结果是我们特别感兴趣的。此外，我们希望解决径向基函数(RBF)逼近适用于所有流形的一些不足之处，即RBF的“本征空间”理论在内插式收敛方面的局限性，以及关于全局支持的RBF的局部逼近能力的问题。此外，作者还希望研究有关约束近似的某些问题。我们提出的研究的一个主要目标是开发更好的方法来表示和分析从空间中的散乱站点或球状物体获得的数据。例如，想象一下在地球上某一特定区域的分散地点获得的温度读数。一个目标是获取温度读数(或烟雾密度读数、臭氧读数等)。通过创建适当的模型(函数)来覆盖整个区域。这样的模型使得在未取样的地点进行“预测性”读数成为可能。此外，在给定区域内采集越来越多的样本，应该会为潜在现象提供一个“更好”的模型。我们建议从数学的角度来研究这些问题。我们注意到，更好地了解数据的表示和分析将对气象学、海洋学和全球定位系统等卫星系统有潜在的好处。

This proposer is particularly interested in representing and analyzing scattered data both on n-dimensional space and on certain compact manifolds such as the m-dimensional sphere by positive definite and related functions. There are now several results related to approximation by positive definite (and related) functions on compact manifolds for which there are no known analogues on n-dimensional Euclidean space for scattered points. For example, quasi-interpolation approximation methods, Marcinkiewicz-Zygmund inequalitiesleading to quadrature formulas associated with scattered points, some approximation rate estimates for multilevel approximation methods and many other results known to be true on spheres do not have good counterparts on n-dimensional Euclidean space. Establishing some corresponding results on these spaces with the idea of better understanding the approximation potential of positive definite functions and their applications is of special interest to us. In addition, we wish to address some of the inadequacies of radial basis function (RBF) approximation applicable to all manifolds; namely, the limitations of the "native space" theory for RBF's in terms of convergence of interpolants, and questions concerning the ability of globally supported RBF's to approximate locally. In addition, the proposer wishes to investigate certain questions about constrained approximation.A major goal of our proposed research is to develop better methods to both represent and analyze data obtained from scattered sites located either in space or from sphere-like objects. For example, imagine temperature readings obtained at scattered sites over a particular region of the earth. One aim is to obtain temperature readings (or smog density readings, ozone readings etc.) over the the entire region by creating an appropriate model(function). Such a model makes "predictive" readings at unsampled sites possible. Also taking an increasing number of samples over a given region should yield a "better" model to the underlying phenomena. We propose the study of such questions from a mathematical point of view. We note that better understanding of the representation and analysis of data would have potential benefits to meteorology, oceanography and satellite-based systems such as the Global Positioning System (GPS).