Mathematical Sciences: Multivariate Approximation

9626319德布尔拟议的研究包括三个主题，都是关于多变量函数的表示和近似的一般领域。它有望在多元样条、多元多项式插值、径向基函数逼近、小波和Gabor展开等领域做出贡献。第一个主题与多项式内插有关。几年前，提出者提出了一种方案，该方案为多维有限离散点集上的任何数据分配一个最小可能次数的多项式插值，并具有各种其他所需的性质。建议建立该格式的误差公式，为其在实验室或三元有限元构造中的应用做准备。第二个主题是移位不变空间逼近理论的应用，这是一种由提出者等人发展起来的理论。一个建议的应用，具有理论性质，但与小波相关，探索可细化空间中函数的光滑性和这种空间提供的逼近阶之间的(令人惊讶的)联系。平移不变空间理论的另一个非常出人意料的应用是在近似来自径向基函数平移范围的散乱数据的领域。该方法基于一个新的转换公式，该公式允许将均匀网格上的许多近似格式推广到一般网格。最后一个主题涉及通过分解/重构技术表示一个或多个变量中的函数。在以往对一般移位不变系统的深入研究中，我们将其应用于新的小波和Gabor系统的构造。特别注意利用过采样系统中固有的冗余所提供的自由。物理问题(包括要由高性能计算解决的问题)的计算机解决方案需要用计算机唯一能理解的语言，即数字来描述问题的各个物理方面。由此产生的数学描述(对车门或人体心脏的描述，对从卫星上拍摄的照片的描述，对空气湿度作为经度、纬度和海拔高度等的函数的描述)。这必然是不准确的，这使得开发有效的表示方法非常重要，即具有可接受的精度和使用不太多的数字的方法。该提案涉及开发这种方法(用于涉及几个参数的描述)以及确定其准确性和效率的手段。

9626319 de Boor The proposed research comprises three topics, all in the general area of representations and approximations of functions of several variables. It is expected to contribute in the areas of Multivariate Splines, Multivariate Polynomial Interpolation, Radial Basis Function Approximation, Wavelets, and Gabor Expansions. The first topic concerns polynomial interpolation. Several years ago, the proposers came up with a scheme that assigns, to any data on any finite discrete pointset in several dimensions, a polynomial interpolant of least possible degree, and with various other desirable properties. It is proposed to develop an error formula for that scheme, in preparation for its application in cubatures or in the construction of trivariate finite elements. The second topic deals with applications of the theory of approximation from shift-invariant spaces, a theory that was developed by the proposers and others. One suggested application, of a theoretical nature, but pertinent to wavelets, explores the (surprising) connection between the smoothness of functions in a refinable space and the approximation orders provided by such a space. Another, very unexpected, application of shift-invariant space theory is in the area of approximation to scattered data from the span of translates of a radial basis function. The suggested approach is based on a new conversion formula that allows the extension of many approximation schemes on uniform grids to general grids. The last topic deals with the representation of functions in one or more variables via decomposition/reconstruction techniques. A thorough understanding reached in prior research on general shift-invariant systems is now used in the constructions of new wavelet and Gabor systems. Special attention is given to exploiting the freedom offered by the redundancy inherent in oversampled systems. Computer solutions of physical problems (including the problems to be att acked by high performance computing) require a description of the various physical aspects of the problem in the only language a computer understands, namely numbers. The resulting mathematical description (of a car door or the human heart, of pictures taken from a satellite, of the moisture content of air as a function of longitude, latitude and height above sea level, etc.) is of necessity inexact, and this makes it very important to develop efficient methods of representation, i.e., methods of acceptable accuracy and using not too many numbers. The proposal concerns the development of such methods (for descriptions involving several parameters) and of means for ascertaining their accuracy and efficiency.