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

DOI:
https://doi.org/10.1090/S0002-9947-1988-0965757-9

MathSciNet review:
965757

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Abstract: For a triangulated -dimensional region in , let denote the vector space of all functions on that, restricted to any simplex in , are given by polynomials of degree at most . 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 in the plane, getting lower bounds on the dimension of for all . For , we prove a conjecture of Strang concerning the generic dimension of the space of splines over a triangulated manifold in . Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0965757-9

Article copyright:
© Copyright 1988
American Mathematical Society