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The approximation order of polysplines

Authors: Ognyan Kounchev and Hermann Render
Journal: Proc. Amer. Math. Soc. 132 (2004), 455-461
MSC (2000): Primary 41A15; Secondary 35J40, 31B30
Published electronically: July 31, 2003
MathSciNet review: 2022369
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Abstract: We show that the scaling spaces defined by the polysplines of order $p$provide approximation order $2p.$ For that purpose we refine the results on one-dimensional approximation order by $L$-splines obtained by de Boor, DeVore, and Ron (1994).

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  • 1. Adams, R., Sobolev Spaces, Academic Press, New York-San Francisco-London, 1975. MR 56:9247
  • 2. de Boor, C., DeVore, R. A., Ron, A., Approximation from shift-invariant subspaces of $L_{2}\left( \mathbb{R} ^{d}\right)$, Trans. Amer. Math. Soc. 341 (1994), pp. 787-806. MR 94d:41028
  • 3. Kounchev, O. I., Definition and basic properties of polysplines, I and II, C. R. Acad. Bulg. Sci., 44 (1991), nos. 7 and 8, pp. 9-11, pp.13-16. MR 93a:41016; MR 92m:41031
  • 4. Kounchev, O. I., Render, H., Multivariate cardinal splines via spherical harmonics. Submitted.
  • 5. Kounchev, O. I., Render, H., The interpolation problem for cardinal splines. Submitted.
  • 6. Kounchev, O. I., Render, H., Symmetry of interpolation polysplines and $L$-splines, Trends in Approximation Theory, K. Kopotun, T. Lyche, and M. Neamtu (eds.), Vanderbilt University Press, Nashville, TN, 2001.
  • 7. Kounchev, O. I., Render, H., Wavelet Analysis of cardinal $L$-splines and construction of multivariate prewavelets, Proceedings of Tenth Approximation Theory Conference (St. Louis, 2001), Innov. Appl. Math. Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 333-353.
  • 8. Micchelli, Ch., Cardinal $L$-splines, In: Studies in Spline Functions and Approximation Theory, Eds. S. Karlin et al., Academic Press, NY, 1976, pp. 203-250. MR 58:1866
  • 9. Kounchev, O. I., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, Boston, 2001. MR 2002h:41001
  • 10. Jetter, K., Plonka G., A survey on $L_{2}$-approximation order from shift-invariant spaces, In: Multivariate Approximation and Applications (N. Dyn, D. Leviatan, D. Levin, and A. Pinkus, eds.), pp. 73-111. Cambridge University Press, 2001. MR 2001m:65005
  • 11. Meyer, Y. Wavelets and Operators, Cambridge University Press, 1992. MR 94f:42001
  • 12. Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971. MR 46:4102

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Additional Information

Ognyan Kounchev
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria

Hermann Render
Affiliation: Institute of Mathematics, University of Duisburg-Essen, Lotharstr. 65, 47048 Duisburg, Germany

Keywords: Cardinal splines, cardinal $L$-splines, polysplines, approximation order of splines, polyharmonic functions, cardinal polysplines.
Received by editor(s): April 5, 2002
Received by editor(s) in revised form: October 2, 2002
Published electronically: July 31, 2003
Communicated by: David Sharp
Article copyright: © Copyright 2003 American Mathematical Society

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