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各有不同，但却有着共同的渊源。此外，许多用于分析基本问题的数学工具都是相似的。在这项工作的大部分基础上，一直存在着一个问题，即开发一种策略，以发展出一种在多个维度上的样条线理论，该理论既在计算上有效，又与产生该主题的一维基础一样准确。关于多元多项式插值的工作源于后来被称为箱样条理论的问题。人们发现了一种令人惊讶的简单而通用的方法，可以为多变量空间中的任何给定的有限点集选择一个好的多项式空间，以便在这些点处进行插值。现在将进行工作，探索这一发现的理论和实践后果。一个长期的目标是构建一个连贯的内插理论，它将在多元数值分析中发挥更重要的作用。目前使用的高维的样条线逼近方法有很多种。每个元素都与一种特定类型的网格相关联，元素沿着该网格连接。这项工作的重点将是逼近阶的研究。更好地理解是什么构成了良好的近似顺序，有望导致构建更好的近似方法。归根结底，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，通常是从给定的一组边界曲线。目前工业工作的主流是一种只对相对平坦的表面起作用的方法。一个直接的目标是更好地理解如何判断一个给定的曲面是否可以由少量的面片很好地表示。

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.