Abstract
It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork ⩾ 20.
Similar content being viewed by others
References
C. de Boor (1973):The quasi-interpolant as a tool in elementary polynomial spline theory. In: Approximation Theory (G. G. Lorentz, ed.). New York: Academic Press, pp. 269–276.
C. de Boor (1975):On bounding spline interpolation. J. Approx. Theory,14:191–203.
K. Höllig (1981):L ∞-boundedness of L 2-projections on splines for a geometric mesh. J. Approx. Theory,33:318–333.
S. Karlin (1986): Total Positivity, Vol. 1. Stanford, California: Stanford University Press.
M. J. Marsden (1974):Quadratic spline interpolation. Bull. Amer. Math. Soc.,80:903–906.
C. A. Micchelli (1976):Cardinal L-splines. In: Studies in Spline Functions and Approximation Theory. New York: Academic Press, pp. 202–250.
C. A. Micchelli (1979):Infinite spline interpolation. In: Approximation in Theorie und Praxis. Ein Symposiumsbericht (G. Meinardus ed.). Mannheim: Bibliographisches Institute, pp. 209–238.
Author information
Authors and Affiliations
Additional information
Communicated by Wolfgang Dahmen.
Rights and permissions
About this article
Cite this article
Jia, R.Q. Spline interpolation at knot averages. Constr. Approx 4, 1–7 (1988). https://doi.org/10.1007/BF02075445
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02075445