Mathematical Sciences: Multivariate Approximation

小行星9626319 拟议的研究包括三个主题，都在一般领域， 多元函数的表示和近似。 是 预计将在多变量样条，多变量 多项式插值，径向基函数逼近，小波， Gabor扩展 第一个主题涉及多项式插值。 几年前，提议者提出了一个方案，将任何 数据的任何有限离散点集在几个维度，多项式 最小可能次数内插，并具有各种其他所需的 特性.建议为该方案建立一个误差公式， 制备其应用在立方或在建设 三元有限元第二个主题涉及 理论的近似从移位不变的空间，一个理论， 由提案人和其他人开发。 一个建议的应用程序， 理论性质，但有关小波，探讨（令人惊讶的） 可加细空间中函数的光滑性与 由这样的空间提供的近似阶。 另一个，非常意外， 移位不变空间理论的应用是在近似于 来自径向基函数的平移跨度的分散数据。的 建议的方法基于一个新的转换公式， 将均匀网格上的许多近似格式推广到一般网格上 网格。 最后一个主题涉及函数在一个或 通过分解/重构技术获得更多变量。 的透彻 对一般平移不变系统的先前研究所达成的理解 目前已被用于构造新的小波和Gabor系统。 特别注意的是利用自由所提供的 过采样系统中固有的冗余。 物理问题的计算机解法（包括待解决的问题） 附加高性能计算）需要描述 问题的各种物理方面的唯一语言计算机 理解，即数字。由此产生的数学描述（ 汽车门或人的心脏，从卫星拍摄的照片， 空气中的水分含量是经度、纬度和海拔高度的函数 海平面等）是不准确的，这使得它非常重要， 制定有效的代表方法，即，可接受的方法 准确性和使用不太多的数字。该提案涉及 这种方法的发展（对于涉及几个 参数）和确定其准确性和效率的方法。

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.