Mathematical Sciences: Smooth Multivariate Piecewise Polynomials; Subdivision Algorithms

该奖项将支持数学研究， 逼近理论和细分算法问题。 近似的工作将处理近似的权力 光滑分段多项式空间的局部结构 二个和三个变量的函数 近期目标是 为了使近似能力的限制精确化， 平滑性要求和底层分区的选择， 近似空间的网格。 底层结构相对简单， 这些问题很难解决，而且不是由一个人引起的-- 维度理论 一个域被细分，一个函数 定义为等于每一片上的某个多项式， 细分（多项式可以随每个片段而变化）。 每一个程度的整体平滑强加给功能 改变了它的逼近能力 的内部相关性 多项式段使得很难计算出 有效的近似程序。 在二维空间中， 例如，使用高度对称的分区（三角形） 得到了紧支函数 已经有了最好的跨度。 工作要在 确定这是否是一个孤立的现象。 细分算法提供了一个有趣的替代方案 到生成平滑曲线和曲面的标准方法。 这样的算法获得曲线或曲面作为 连续细化的分段线性元素的序列， 每一个都是从它的前身上“削”下来的。 它们是计算机辅助领域的基本工具 几何设计 人们可以控制限制的形状 通过控制初始虚线的选择， example. 因此，人们可以设计特定的形状， 达到指定的设计特点，通过这些 算法 目前尚不完全清楚的是， 细分算法总是收敛的。 正如所 预计，有关表面的问题将更加困难 而不是曲线。 即使在后一种情况下， 示出了对于所有外观而言 平滑（在显示器上）实际上是分形。

This award will support mathematical research on problems of approximation theory and subdivision algorithms. Work on approximations will deal with the approximation power and local structure of spaces of smooth piecewise polynomial functions of two and three variables. The immediate goal is to make precise the limitations put on approximation power by smoothness demands and choice of the underlying partition or mesh of the approximating space. The underlying structure is relatively simple although the problems are difficult and not motivated by one- dimensional theory. A domain is subdivided and a function is defined to be equal to some polynomial on each piece of the subdivision (the polynomials can change with each piece). Each degree of overall smoothness forced on the function changes its approximating power. The interdependency of the polynomial pieces makes it difficult to come up with effective approximation procedures. In two-dimensions for example, use of highly symmetrical partitions (triangles) results in the theorem that the compactly supported functions already have best-possible span. Work is to be done in establishing whether or not this is an isolated phenomenon. Subdivision algorithms provide an intriguing alternative to standard ways of generating smooth curves and surfaces. Such algorithms obtain a curve or surface as the limit of a sequence of successively refined piecewise linear elements, each obtained by "whittling" pieces off its predecessor. They are fundamental tools in the area of computer aided geometric design. One can control the shape of the limiting curve by controlling the choice of initial broken lines, for example. Thus one may be able to design specific shapes and arrive at specified design characteristics through these algorithms. What is not completely clear is whether or not the subdivision algorithms always converge. As might be expected, the problems concerning surfaces are more difficult than those for curves. Yet even in the latter case, examples show that limiting curves which for all appearances are smooth (on a monitor) are actually fractals.