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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homology of smooth splines: generic triangulations and a conjecture of Strang


Author: Louis J. Billera
Journal: Trans. Amer. Math. Soc. 310 (1988), 325-340
MSC: Primary 41A15; Secondary 65D07
MathSciNet review: 965757
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Abstract: For $ \Delta $ a triangulated $ d$-dimensional region in $ {{\mathbf{R}}^d}$, let $ S_m^r(\Delta )$ denote the vector space of all $ {C^r}$ functions $ F$ on $ \Delta $ that, restricted to any simplex in $ \Delta $, are given by polynomials of degree at most $ m$. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds $ \Delta $ in the plane, getting lower bounds on the dimension of $ S{}_m^r(\Delta )$ for all $ r$. For $ r = 1$, we prove a conjecture of Strang concerning the generic dimension of the space of $ {C^1}$ splines over a triangulated manifold in $ {{\mathbf{R}}^2}$. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965757-9
PII: S 0002-9947(1988)0965757-9
Article copyright: © Copyright 1988 American Mathematical Society