Algebraic methods in approximation theory
For researchers in approximation theory and geometric modeling, this provides a consolidated overview of algebraic techniques, but it is a survey of existing methods rather than a novel contribution.
This survey reviews algebraic methods (homology, graded algebra, localization, inverse systems) for studying splines, highlighting progress such as a formula for the third coefficient of the dimension polynomial of spline spaces in high degree.
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in R^k, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree, much of which builds on pioneering work of Schumaker, Alfeld-Schumaker, and Billera. The objects appearing here may be computed using the Macaulay2 software system.