The approximation order of box spline spaces

Authors:
A. Ron and N. Sivakumar

Journal:
Proc. Amer. Math. Soc. **117** (1993), 473-482

MSC:
Primary 41A15; Secondary 41A25, 41A63

DOI:
https://doi.org/10.1090/S0002-9939-1993-1110553-2

MathSciNet review:
1110553

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a box spline associated with an arbitrary set of directions and suppose that is the space spanned by the integer translates of . In this note, the subspace of all polynomials in is shown to be the joint kernal of a certain collection of homogeneous differential operators with constant coefficients. The approximation order from the dilates of to smooth functions is thereby characterized. This extends a well-known result of de Boor and Höllig (-*splines from parallelepipeds*, J. Analyse Math. **42** (1982/83), 99-115), on box splines with integral direction sets.

The argument used is based on a new relation, valid for any compactly supported distribution , between the semidiscrete convolution and the distributional convolution .

**[B]**Carl de Boor,*The polynomials in the linear span of integer translates of a compactly supported function*, Constr. Approx.**3**(1987), no. 2, 199–208. MR**889555**, https://doi.org/10.1007/BF01890564**[BAR]**Asher Ben-Artzi and Amos Ron,*Translates of exponential box splines and their related spaces*, Trans. Amer. Math. Soc.**309**(1988), no. 2, 683–710. MR**961608**, https://doi.org/10.1090/S0002-9947-1988-0961608-7**[BH]**C. de Boor and K. Höllig,*𝐵-splines from parallelepipeds*, J. Analyse Math.**42**(1982/83), 99–115. MR**729403**, https://doi.org/10.1007/BF02786872**[BHR]**C. de Boor, K. Höllig, and S. D. Riemenschneider,*Box splines*, Springer-Verlag, Berlin and New York, 1992.**[BR1]**Carl de Boor and Amos Ron,*Polynomial ideals and multivariate splines*, Multivariate approximation theory, IV (Oberwolfach, 1989) Internat. Ser. Numer. Math., vol. 90, Birkhäuser, Basel, 1989, pp. 31–40. MR**1034294****[BR2]**-,*The exponentials in the span of the integer translates of a compactly supported function*:*approximation order and quasi-interpolation*, J. London Math. Soc. (2) (to appear).**[C]**Charles K. Chui,*Multivariate splines*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR**1033490****[DM]**Wolfgang Dahmen and Charles A. Micchelli,*Translates of multivariate splines*, Linear Algebra Appl.**52/53**(1983), 217–234. MR**709352**, https://doi.org/10.1016/0024-3795(83)80015-9**[F]**F. G. Friedlander,*Introduction to the theory of distributions*, Cambridge University Press, Cambridge, 1982. MR**779092****[JR]**K. Jetter and S. D. Riemenschneider,*Cardinal interpolation with box splines on submodules of*, Approximation Theory V (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, New York, 1986, pp. 403-406.**[R]**Amos Ron,*A characterization of the approximation order of multivariate spline spaces*, Studia Math.**98**(1991), no. 1, 73–90. MR**1110099**, https://doi.org/10.4064/sm-98-1-73-90**[SF]**G. Strang and G. Fix,*A Fourier analysis of the finite element variational method*, C.I.M.E. II Cilo 1971, Constructive Aspects of Functional Analysis (G. Geymonet, ed.), 1973, pp. 793-840.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1110553-2

Keywords:
Box splines,
polynomials,
multivariate splines,
approximation order

Article copyright:
© Copyright 1993
American Mathematical Society