Title
Multivariate Polysplines: Applications To Numerical And Wavelet Analysis
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Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions.Multivariate polysplines have applications in the design of surfaces and 'smoothing' that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than wellestablished methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature. Part 1 assumes no special knowledge of partial differential equations and is intended as a graduate level introduction to the topic Part 2 develops the theory of cardinal Polysplines, which is a natural generalization of Schoenberg's beautiful onedimensional theory of cardinal splines Part 3 constructs a wavelet analysis using cardinal Polysplines. The results parallel those found by Chui for the onedimensional case Part 4 considers the ultimate generalization of Polysplines on manifolds, for a wide class of higherorder elliptic operators and satisfying a Holladay variational property
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