The approximation order of box spline spaces
Authors:
A. Ron and N. Sivakumar
Journal:
Proc. Amer. Math. Soc. 117 (1993), 473482
MSC:
Primary 41A15; Secondary 41A25, 41A63
MathSciNet review:
1110553
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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 wellknown result of de Boor and Höllig (splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99115), 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 .
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de Boor, The polynomials in the linear span of integer translates
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(1987), no. 2, 199–208. MR 889555
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BenArtzi and Amos
Ron, Translates of exponential box splines
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C. de Boor, K. Höllig, and S. D. Riemenschneider, Box splines, SpringerVerlag, Berlin and New York, 1992.
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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. 403406.
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Amos
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 [B]
 C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), 199208. MR 889555 (88e:41054)
 [BAR]
 A. BenArtzi and A. Ron, Translates of exponential box splines and their related spaces, Trans. Amer. Math. Soc. 309 (1988), 683710. MR 961608 (89m:41008)
 [BH]
 C. de Boor and K. Höllig, splines from parallelepipeds, J. Analyse Math. 42 (1982/3), 99115. MR 729403 (86d:41008)
 [BHR]
 C. de Boor, K. Höllig, and S. D. Riemenschneider, Box splines, SpringerVerlag, Berlin and New York, 1992.
 [BR1]
 C. de Boor and A. Ron, Polynomial ideals and multivariate splines, Multivariate Approximation Theory V (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1990, pp. 3140. MR 1034294 (91g:41009)
 [BR2]
 , The exponentials in the span of the integer translates of a compactly supported function: approximation order and quasiinterpolation, J. London Math. Soc. (2) (to appear).
 [C]
 C. K. Chui, Multivariate splines, CBMSNSF Regional Conf. Ser. Appl. Math., vol. 54, SIAM, Philadelphia, 1988. MR 1033490 (92e:41009)
 [DM]
 W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/3 (1983), 217234. MR 709352 (85e:41033)
 [F]
 H. G. Friedlander, Introduction to the theory of distributions, Cambridge Univ. Press, London and New York, 1982. MR 779092 (86h:46002)
 [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. 403406.
 [R]
 A. Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1991), 7390. MR 1110099 (92g:41017)
 [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. 793840.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311105532
PII:
S 00029939(1993)11105532
Keywords:
Box splines,
polynomials,
multivariate splines,
approximation order
Article copyright:
© Copyright 1993
American Mathematical Society
