Mathematical Sciences: Theory and Applications of Multivariate Splines

这个项目代表了研究人员对多元(多项式)样条线的理论、应用和计算方面的研究的继续。该项目的理论方面可以看作是应用分析，在应用中强调计算和实施的效率。确定多元样条空间的逼近能力是所要解决的问题之一。一旦已知，下一步是识别具有相同逼近能力的低得多的维子空间，以便此子空间的特定特征可以有效地构造具有最佳逼近阶的逼近。另一个问题是开发能保证最佳逼近阶的有效方案。由于插值法有时很难实现，在某些情况下甚至可能不理想，因此研究拟插值法的构造也很重要。最近，研究人员刻画了所有由紧支撑函数诱导的网格数据的拟插值项。由此，人们应该能够研究具有“最小支撑度”的准内插式。对于散乱数据，该问题将通过顶点样条线来解决。多元样条函数最重要的应用之一是构造具有一定保形特性的逼近函数。这个项目的一部分是应用在前一个项目中开发的理论来研究该方法对形状保持问题的影响。最后，进一步研究了重要子空间的局部基的维数和构造的基本问题。

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.