Approximations of Functions from Scattered Data: Theory and Applications

这个提议者特别感兴趣的是表示和分析分散的数据都在n维空间和某些紧凑的流形，如m维球的正定和相关的功能。现在有几个结果有关的逼近正定（和相关）的功能，有没有已知的类似物在n维欧氏空间的散点紧流形。例如，拟插值逼近方法，Marcinkiewicz-Zygmund不等式导致与散乱点相关的求积公式，多层逼近方法的一些逼近率估计以及许多其他已知在球面上为真的结果在n维欧氏空间上没有很好的对应物。为了更好地理解正定函数的逼近势及其应用，在这些空间上建立相应的结果是我们特别感兴趣的。此外，我们希望解决一些不足之处的径向基函数（RBF）近似适用于所有流形，即，“本地空间”理论的RBF的插值收敛方面的局限性，和问题的能力，全球支持的RBF的局部近似。此外，本文还对约束近似的一些问题进行了研究，主要目的是寻找更好的方法来表示和分析空间或球形物体上的散乱数据。例如，想象一下在地球上某个特定区域的分散地点获得的温度读数。一个目的是获得温度读数（或烟雾密度读数，臭氧读数等）。通过创建适当的模型（函数），在整个区域上进行。这样的模型使得在未取样地点的“预测”读数成为可能。此外，在一个给定的区域采取越来越多的样本应该产生一个“更好的”模型的基本现象。我们建议从数学的角度来研究这些问题。我们注意到，更好地理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产生潜在的好处。

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).