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，但有共同的来源。 此外，许多用于分析基本问题的数学工具都是相似的。 这项工作的大部分工作的基础是开发一种策略来开发多维样条理论的问题，该策略在计算上既有效又与该主题产生的一维基础一样准确。 多元多项式插值的工作源于后来被称为箱形样条理论的问题。 已经发现了一种令人惊讶的简单且通用的方法，用于为多个变量的空间中的任何给定的有限点集选择在这些点处插值的良好多项式空间。 现在将开展工作，探索这一发现的理论和实践影响。 长期目标是构建一种连贯的插值理论，该理论将在多元数值分析中发挥更重要的作用。 目前使用的高维样条逼近方法有很多。 每个都与某种类型的网格相关联，元素沿着该网格连接。 这项工作的重点是近似阶数。 更好地理解如何形成良好的近似阶预计将有助于构建更好的近似方法。最终，人们希望开发出一种现有的各种理论和技术的统一。 这项工作的一个特别重要的应用领域是提供表面的数学模型，通常来自一组给定的边界曲线。 此时工业工作的主要方法是仅适用于相对平坦的表面的方法。 一个直接的目标是更好地理解如何判断给定的表面是否可以用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.