Mathematical Sciences: Theory and Applications of Multivariate Splines

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This project represents the continuation of the investigators' research into the theory, applications, and computational aspects of multivariate (polynomial) splines. The theoretical aspects of this project may be considered as applied analysis and efficiency in computations and implementation is emphasized in the applications. One of the problems addressed is to determine the approximation power of a multivariate spline space. Once it is known, the next step is to identify a much lower dimensional subspace with the same approximation power so that specific features of this subspace can be used to efficiently construct approximates with optimal order of approximation. Another problem is to develop efficient schemes that guarantee the optimal order of approximation. Since interpolation is sometimes expensive to attain, and in some situations may not even be desirable, it is also important to study the constrution of quasi-interpolants. Recently the investigators have characterized all quasi-interpolants induced by a compactly supported function to gridded data. From this, one should be able to study quasi-interpolants with "minimum support". For scattered data, the problem will be approached via vertex splines. One of the most important applications of multivariate splines is the construction of approximants that have certain shape preserving characteristics. Part of this project is to apply the theory developed during the previous project to investigate the implications of that approach to shape preserving problems. Finally, the basic problems on dimension and construction of local bases of the important subspaces will be investigated further.