Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-03T00:16:34.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Cardinal interpolation with shifted three-directional box splines†

Published online by Cambridge University Press:  14 November 2011

Peter Binev  and
Kurt Jetter
Peter Binev
Affiliation:
Fachbereich Mathematik, Universität Duisburg, D-4100 Duisburg 1, Germany
Kurt Jetter
Affiliation:
Fachbereich Mathematik, Universität Duisburg, D-4100 Duisburg 1, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

The question of “correctness” of cardinal interpolation with shifted three-directional box splines is solved for arbitrary orders of the directional vectors. It is shown that the corresponding symbol can be viewed as a collection of curves with certain properties (convexity, increasing argument, etc.) which are investigated in detail. The method of proof involves an induction argument which is based on properties of the exponential Euler splines (studied in [6]).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 205 - 220
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1de Boor, C., Höllig, K. and Riemenschneider, S. D.. Bivariate cardinal interpolation by splines on a three-direction mesh. Illinois J. Math. 29 (1985), 533–566.CrossRefGoogle Scholar
2Chui, C. K.. Multivariate Splines, CBMS-NSF Reg. Conf. Series in Appl. Math. 54 (Philadelphia; SIAM, 1988).CrossRefGoogle Scholar
3Chui, C. K., Stöckier, J. and Ward, J. D.. Invertibility of shifted box spline interpolation operators. SIAM J. Math. Anal. 22 (1991), 543–553.CrossRefGoogle Scholar
4Chui, C. K., Stöckier, J. and Ward, J. D.. Singularity of cardinal interpolation with shifted box splines. (CAT Report 217, Texas A & M University, May, 1990).Google Scholar
5Jetter, K., A short survey on cardinal interpolation by box splines. In Topics in Multivariate Approximation, eds Chui, C. K., Schumaker, L. L. and Utreras, F. I., 125–139 (New York: Academic Press, 1987).CrossRefGoogle Scholar
6Jetter, K., Riemenschneider, S. D. and Sivakumar, N.. Schoenberg's exponential Euler spline curves. Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 21–33.CrossRefGoogle Scholar
7Sivakumar, N.. Studies in box splines (Ph.D. Thesis, University of Alberta, Edmonton, 1990).Google Scholar
8Sivakumar, N.. On bivariate cardinal interpolation by shifted splines on a three-direction mesh. J. Approx. Theory 61 (1990), 178–193.CrossRefGoogle Scholar
9Stöckier, J.. Cardinal interpolation with translates of shifted bivariate box-splines. In Mathematical Methods in Computer Aided Geometric Design, eds Lyche, T. and Schumaker, L. L., 583–592 (Boston: Academic Press, 1989).Google Scholar