Modulation Splines

这项建议的核心是引入和分析一类新的样条函数，这是自1980年代中期S引入多项式和指数盒样条法以来的第一次此类努力。这类新的样条法是在“调制样条法”的建议中提出的，它与所有经典的样条法范例有很大的不同：虽然调制样条法是“样条法”，即在多面体区域上光滑紧支撑的分段解析函数，但即使在一维上它们也是新的。在短期内，建议发展调制样条法的基本理论和基本性质，重点是2D构造。从长远来看，调制样条法将作为一种新的分层各向异性多变量数据表示方法的基础，这种表示方法不同于主流的傅立叶和小波表示法，并且是对它们的补充。与所有的样条理论和构造一样，该项目位于数学分析和计算科学的交界处。然而，该项目的推动力来自非交换代数中的一个主题，即麦克唐纳多项式，这是一个与李代数和群表示相关的主题。因此，该项目进一步证明了数学科学中内在联系的无限潜力，并有望在分析、计算和代数之间提供一条相互促进的渠道。相对于NSF更广泛的优点标准，该项目的核心优势是研究领域的内在意义：数据表示法，特别是样条法近似，是科学中的关键学科，这些领域的进展可能在广泛的学科范围内产生广泛的乘数效应。事实上，“样条函数”的理论和实践是数学界对科学技术最重大的贡献之一。在汽车和飞机的计算机辅助设计和制造中，在打印机排版的生产中，在自动制图中，在电影制作中，在许多其他领域，样条线已经成为不可或缺的工具，这些工具往往隐藏在复杂的软件包的核心中。除了直接用于曲线和曲面的表示外，在一个或多个变量中，样条是小波表示的首选主干，也是计算机辅助几何设计中平滑细分算法的主流选择。对样条线数学研究的影响的认识在本世纪初达到顶峰，美国总统授予卡尔·德布尔科学奖章。

This proposal centers on the introduction and analysis of a new class of spline functions, the first such endeavor since the introduction of polynomial and exponential box splines in the mid 1980's. Coined in the proposal "modulation splines", the novel class represents a dramatic departure from all classical spline paradigms: While modulation splines are "splines", i.e., smooth compactly supported piecewise-analytic functions over polyhedral domains, they are new even in one dimension.In the short term, it is proposed to develop the basic theory and fundamental properties of modulation splines, with emphasis on 2D constructions. In the longer term, it is envisioned that modulation splines will serve as the backbone of a new hierarchical anisotropic multivariate data representation methodology, representation that is different and complementary to the prevailing Fourier and wavelet ones. As is the case with all spline theories and constructions, the project lies at the interface between mathematical analysis and computational science. The impetus for the project, however, comes from a topic in non-commutative algebra known as Macdonald polynomials, a topic that is related to Lie algebras, and to group representations. As such, the project provides further evidence to the unlimited potential in the intraconnectivity within mathematical science, and is anticipated to provide a channel of cross-fertilization among analysis, computation, and algebra.The project's core strength vis-a-vis NSF's broader merit criteria is the intrinsic significance of the research area: data representation in general, and spline approximation in particular, are critical disciplines in science, and progress in these areas may have a widespread multiplier effect in a broad range of disciplines. In fact, the theory and practice of "spline functions" stands out as one of the most significant contributions of the mathematical community to science and technology. Splines have become indispensable tools in computer-aided design and manufacturing of cars and airplanes, in the production of printers' typesets, in automated cartography, in the production of movies, and in many other areas, often concealed at the core of elaborate software packages. In addition to their direct utility for the representation of curves and surfaces, splines, in one as well as several variables, are the preferred backbone for the wavelet representation, and are the prevailing choice for smoothing subdivision algorithms in computer-aided geometric design. The recognition of the impact of the mathematical research of splines culminated earlier in this decade in the awarding of a Medal of Science to Carl de Boor by the U.S. president.