Algebraic Methods for Supersmooth Spline Spaces

TL;DR

This paper develops algebraic methods to analyze supersmooth spline spaces on polyhedral complexes, providing dimension bounds and exact calculations for planar triangulations with enhanced smoothness conditions.

Contribution

It introduces a generalized algebraic framework for supersmooth splines, extending existing complexes to incorporate additional smoothness constraints and deriving dimension bounds.

Findings

A generalized algebraic complex for supersmooth splines is constructed.

A lower bound for the dimension of supersmooth spline spaces on planar triangulations is established.

The lower bound matches the actual dimension in high degree cases.

Abstract

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera-Schenck-Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.