Mathematical Sciences: Multivariate Approximation

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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.