Algebraic methods in approximation theory

TL;DR

This survey reviews algebraic techniques like homology, graded algebra, localization, and inverse systems used to analyze splines, highlighting recent progress and computational tools in approximation theory.

Contribution

It introduces and illustrates four algebraic methods for studying splines, providing concrete examples and recent advancements in the field.

Findings

A formula for the third coefficient of the spline dimension polynomial

Progress in understanding spline spaces using algebraic methods

Computational implementation with Macaulay2

Abstract

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.