Mathematical Sciences: Multivariate Polynomial Interpolationand Spline Approximation

多元多项式的数学理论 插值和多元样条近似不同之处在于 内容和目标，但共享一个共同的来源。 另外很多 用于分析基本问题的数学工具有 相似 这项工作的主要问题是 发展战略，发展样条理论， 几个维度，既计算有效， 准确的作为一维的基础， 升起。 多变量多项式插值的工作来源于 箱样条理论中的问题 一 令人惊讶的简单和通用的方法来选择，对于任何 给定多变量空间中的有限点集， 在这些点上插值的良好多项式空间，已经被 发现了 现在将进行工作，探索理论和 这一发现的实际后果。 一个长期目标 是建立一个连贯的插值理论， 将在多元数值计算中发挥更重要的作用 分析. 在高等数学中，样条逼近有多种方法， 目前正在使用的尺寸。 每一个都与一个特定的 元素沿其连接的网格沿着类型。 的焦点 这个工作将是近似阶的工作。 更好的 理解什么是好的近似阶是 预期将导致更好的近似构造 方法.最终，人们希望发展一种统一的 现存的各种理论和技术。 这项工作的一个特别重要的应用领域是 在提供表面的数学模型时，通常从给定的 一组边界曲线。 这一时期工业工作的支柱 时间是一种仅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了解如何 可以判断给定的表面是否可以很好地由 少量的补丁。

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.