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A lower bound for the dimension of tetrahedral splines in large degree

Michael DiPasquale, Nelly Villamizar Orcid Logo

Constructive Approximation

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1007/s00365-023-09625-5

Abstract

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable at...

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Published in: Constructive Approximation
ISSN: 0176-4276 1432-0940
Published: Springer Science and Business Media LLC 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63205
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Abstract: Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.
Keywords: Trivariate spline spaces, tetrahedral partitions, dimension of spline spaces.
College: Faculty of Science and Engineering
Funders: Swansea University. EPSRC.

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