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Algebraic methods to study the dimension of supersmooth spline spaces

DeepeshToshniwal, Nelly Villamizar Orcid Logo

Advances in Applied Mathematics, Volume: 142

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1016/j.aam.2022.102412

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...

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Published in: Advances in Applied Mathematics
ISSN: 0196-8858
Published: Elsevier 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa60699
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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.
Keywords: Spline functions; Superspline spaces on triangulations; Dimension of spline spaces; Supersmoothness; Intrisic supersmoothness
College: Faculty of Science and Engineering
Funders: EPSRC (New Investigator Award/EP/V012835/1); Dutch Research Council (Veni research programme/212.150)

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