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.
