Mathematical Sciences: Multivariate Polynomial Interpolationand Spline Approximation

多元多项式插值和多元样条逼近的数学理论在内容和目标上有所不同，但有一个共同的来源。此外，许多用于分析基本问题的数学工具是相似的。这项工作的基础是发展一种策略，以发展一种在几个维度上的样条理论，这种理论在计算上是有效的，并且与该主题产生的一维基础一样准确。多元多项式插值的研究源于后来被称为框样条理论的问题。对于在多变量空间中任意给定的有限点集，我们发现了一种非常简单和通用的选择方法，在这些点上找到了一个好的多项式插值空间。现在的工作是探索这一发现的理论和实践后果。一个长期的目标是建立一个连贯的插值理论，它将在多元数值分析中发挥更重要的作用。目前有许多高维的样条逼近方法。每个都与特定类型的网格相关联，元素沿着网格连接。这项工作的重点将是近似阶。更好地理解是什么促成了一个好的近似顺序，有望导致更好的近似方法的构建。最终，人们希望将现有的各种理论和技术统一起来。这项工作的一个特别重要的应用领域是提供曲面的数学模型，通常来自一组给定的边界曲线。在这个时候，工业工作的主流是一种只适用于相对平坦表面的方法。一个直接的目标是更好地理解如何判断一个给定的表面是否可以由少量的斑块很好地表示。

The mathematical theories of multivariate polynomial interpolation and multivariate spline approximation differ in content and goals, yet share a common source. In addition, many of the mathematical tools used to analyze basic questions are similar. Underlying much of this work has been the problem of developing a strategy for developing a theory of splines in several dimensions which is both computationally effective and as accurate as the one dimensional basis from which the subject arose. Work on multivariate polynomial interpolation derives from problems in what became known as box spline theory. A surprisingly simple and general method for choosing, for any given finite set of points in a space of several variables, a good polynomial space for interpolation at those points, has been discovered. Work will now be done exploring the theoretical and practical ramifications of this discovery. A long-term objective is to construct a coherent theory of interpolation, one which will play a more important role in multivariate numerical analysis. There are many approaches to spline approximation in higher dimensions currently in use. Each is associated with a certain type of mesh along which the elements are joined. The focus of this work will be that of approximation order. A better understanding of what makes for a good approximation order is expected to lead to the construction of better approximation methods. Ultimately, one would like to develop a unification of the various theories and techniques now extant. A particularly important area of application of this work is in providing mathematical models of surfaces, often from a given set of bounding curves. The mainstay of industrial work at this time is a method which only works for relatively flat surfaces. One immediate goal is to obtain a better understanding of how one can tell whether a given surface can be well represented by a small number of patches.