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

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- by Louis J. Billera
- Trans. Amer. Math. Soc.
**310**(1988), 325-340 - DOI: https://doi.org/10.1090/S0002-9947-1988-0965757-9
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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.## References

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## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- 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