Partitions of unity and approximation

de Boor, C.; DeVore, R.

doi:10.1090/S0002-9939-1985-0776207-2

Partitions of unity and approximation
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by C. de Boor and R. DeVore PDF

Proc. Amer. Math. Soc. 93 (1985), 705-709 Request permission

Abstract:

We show that for certain translation invariant spaces $S$, a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces ${S_h}$ is that $S$ contain a stable and locally supported partition of unity. These results have been motivated by recent work on approximation by multivariate piecewise polynomials on regular meshes.

References

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Approximation order from smooth pp functions

C. de Boor and K. Höllig, Bivariate box splines and smooth pp functions on a three direction mesh, J. Comput. Appl. Math. 9 (1983), no. 1, 13–28. MR 702228, DOI 10.1016/0377-0427(83)90025-0
C. de Boor and K. Höllig, Approximation order from bivariate $C^{1}$-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), no. 4, 649–655. MR 687634, DOI 10.1090/S0002-9939-1983-0687634-4

On the optimal order approximation rates for criss-cross finite element spaces

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