Partitions of unity and approximation

Authors:
C. de Boor and R. DeVore

Journal:
Proc. Amer. Math. Soc. **93** (1985), 705-709

MSC:
Primary 41A15

MathSciNet review:
776207

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Abstract: We show that for certain translation invariant spaces , a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces is that 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.

**[BD]**C. de Boor and R. DeVore,*Approximation by smooth multivariate splines*, Trans. Amer. Math. Soc.**276**(1983), no. 2, 775–788. MR**688977**, 10.1090/S0002-9947-1983-0688977-5**[BDH]**C. de Boor, R. DeVore and K. Höllig,*Approximation order from smooth pp functions*, Approximation Theory. IV (C. Chui, L. L. Schumaker and J. Ward, editors), Academic Press, New York, 1983, pp. 353-357.**[BH]**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**, 10.1016/0377-0427(83)90025-0**[BH]**C. de Boor and K. Höllig,*Approximation order from bivariate 𝐶¹-cubics: a counterexample*, Proc. Amer. Math. Soc.**87**(1983), no. 4, 649–655. MR**687634**, 10.1090/S0002-9939-1983-0687634-4**[DM]**W. Dahmen and C. Micchelli,*On the optimal order approximation rates for criss-cross finite element spaces*, 1983.**[J]**Rong-ging Jia,*Approximation by smooth bivariate splines on a three direction mesh*, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 539–545. MR**754389**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1985-0776207-2

Keywords:
Degree of approximation,
partition of unity,
piecewise polynomial

Article copyright:
© Copyright 1985
American Mathematical Society