Modulation Splines

本提案的重点是引入和分析一类新的样条函数，这是自20世纪80年代中期引入多项式和指数样条以来的第一次这样的努力。在“调制样条”的提议中，新类代表了对所有经典样条范式的巨大背离：虽然调制样条是“样条”，即多面体域上的光滑紧支持分段解析函数，但它们即使在一维中也是新的。在短期内，建议发展调制样条的基本理论和基本性质，重点是二维结构。从长远来看，预计调制样条将作为一种新的分层各向异性多元数据表示方法的支柱，这种表示与流行的傅立叶和小波表示不同且互补。与所有样条理论和结构一样，该项目位于数学分析和计算科学之间的界面。然而，该项目的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内部联系的无限潜力，并有望在分析、计算和代数之间提供一个交叉受精的渠道。相对于美国国家科学基金会更广泛的优点标准，该项目的核心优势在于研究领域的内在意义：数据表示，特别是样条近似，是科学中的关键学科，这些领域的进展可能会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要的贡献之一。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，以及在许多其他领域，样条已经成为不可或缺的工具，通常隐藏在精心制作的软件包的核心。除了直接用于曲线和曲面的表示之外，样条，在一个或几个变量中，是小波表示的首选主干，并且是计算机辅助几何设计中平滑细分算法的普遍选择。本世纪初，美国总统向卡尔·德·布尔（Carl de Boor）颁发了一枚科学奖章，从而使人们认识到样条数学研究的影响达到了顶峰。

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.