Hostname: page-component-54dcc4c588-ff9ft Total loading time: 0 Render date: 2025-09-11T21:53:10.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Cardinal interpolation by Dm-splines

Published online by Cambridge University Press:  14 November 2011

T. N. T. Goodman  and
S. L. Lee
T. N. T. Goodman
Affiliation:
Department of Mathematical Sciences, University of DundeePenang, Malaysia
S. L. Lee
Affiliation:
School of Mathematical Sciences, University of Science of Malaysia, Penang, Malaysia
Get access Rights & Permissions [Opens in a new window]

Synopsis

Duchon (1978) considered interpolation in ℝn by “Dm-splines”, which are interpolating functions having, in a sense, minimum energy. The purpose of this paper is to consider the analogous interpolation at the lattice of points in ℝn with integer co-ordinates, generalising aspects of Schoenberg's (1973) theory of cardinal spline interpolation. Following Schoenberg, we prove that higher order “basic” splines can be written as convolutions of lower order ones, using a new notion of convolution due to Jones (1982).

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 1-2 , 1983 , pp. 149 - 161
Copyright
Copyright © Royal Society of Edinburgh 1983

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.)

Article purchase

Temporarily unavailable

References

1Aronszajn, N.. Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950), 337–401.CrossRefGoogle Scholar
2Neny, J. and Lions, J. L.. Les espaces du type de Beppo Levi. Ann. Inst. Fourier, Grenoble 5 (1954), 305–370.Google Scholar
3Duchon, J.. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Anal. Numér. 10 (no. 12) (1976), 5–12.Google Scholar
4Duchon, J.. Sur l'erreur d'interpolation des fonctions de plusieurs variables par les Dm-splines. RAIRO Anal. Numér. 12 (No. 4) (1978), 325–334.CrossRefGoogle Scholar
5Golomb, M. and Weinberger, H. F.. Optimal approximation and error bounds. In On Numerical Approximation, pp. 117–190 (Langer, R. E., Ed.) (Madison: University of Winconsin Press, 1959).Google Scholar
6Jones, D. S.. Infinite integrals and convolution. Proc. Roy. Soc. London Ser. A 371 (1980), 479–508.Google Scholar
7Jones, D. S.. Generalised functions and their convolution. Proc. Roy. Soc. Edinburgh Sect. A 91 (1982), 213–233.CrossRefGoogle Scholar
8Meinguet, J.. Basic mathematical aspects of surface spline interpolation. Numerische Integration (Tagung Math. Forschungsinst., Oberwolfach, 1978), pp. 211–220. Internat. Ser. Numer. Math. 45 (Basel: Birkhäuser, 1979).CrossRefGoogle Scholar
9Schoenberg, I. J.. On spline functions. In Inequalities, pp. 255–291 (Shisha, O., Ed.) (New York: Academic Press, 1967).Google Scholar
10Schoenberg, I. J.. Cardinal spline interpolation. CBMS Vol. 12, (Philadelphia: SIAM, 1973).CrossRefGoogle Scholar
11Schwartz, L.. Thérie des distributions (Paris: Hermann, 1972).Google Scholar
12Schwartz, L.. Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (Noyaux reproduisants). J. Anal. Math. 13 (1964), 115–256.CrossRefGoogle Scholar