Mathematical Sciences: Theory and Applications of Multivariate Splines

该项目代表了 研究人员对理论、应用和 多变量（多项式）样条的计算方面。 该项目的理论方面可以被认为是 应用分析和计算效率， 在应用程序中强调执行。 之一 解决的问题是确定近似功率的 多元样条空间 一旦知道了，下一步就是 来识别一个低维度的子空间， 近似能力使得这个子空间的特定特征 可以用来有效地构建近似与最佳 近似的顺序。 另一个问题是发展 有效的计划，保证最佳顺序 近似 由于插值有时很昂贵， 在某些情况下，甚至可能是不可取的， 对研究拟插值的逼近也很重要。 最近，调查人员对所有 由紧支函数诱导的拟插值 网格数据 从这一点，一个人应该能够研究 具有“最小支持度”的拟插值。 对于分散的数据， 该问题将通过顶点样条来处理。 之一 多元样条最重要的应用是 保形逼近的构造 特色 这个项目的一部分是应用这个理论 在上一个项目期间开发，以调查 这种方法对形状保持问题的影响。 最后，讨论了关于尺度和构造的基本问题， 重要子空间的局部基将被研究 进一步.

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.