Mathematical Sciences: Multivariate Approximation

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9224748 deBoor This project focuses on optimal approximation and interpolation. The work expands on recent research on approximation orders in the quadratic and supremum norms. It is expected that the approximation order of the intermediate Lebesgue spaces will remain constant. In addition, work will be done on finding the dimension of box spline approximation spaces in three dimensions. Efforts will be made in seeking a Fourier analysis approach to solve approximation order questions in seeking a more general approach applicable to any scale, whether local or not. Further research will be carried out on shift-invariant spaces using shifts of radial functions as suitable approximants or restricted families of functions. New approaches to the wavelet construct, the multiresolution analysis, has led to an expanded nonstationary wavelet which may have practical benefit. Related to this will be a close study of the stability of shifts of solutions of dilation equations in terms of their Fourier transforms. Continuing research on approximation of surfaces and the problems of blending patches of surfaces will continue. At the present time, blended patches are only satisfactory when the given surfaces are relatively flat. Better understanding is needed to tell whether a given surface can be well represented by a few blended patches. Approximation theory combines techniques of special functions, numerical analysis and functional analysis to analyze complex problems in which closed form or simple solutions are not feasible. By its very nature, such research carries an implicit goal involving application to questions of the physical world, although some of the work can be highly theoretical. ***