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Optimal interpolation

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 506)

Abstract

The classical interpolation problem is considered of estimating a function of one variable, f(.), given a number of function values f(xi), i=1,2,...,m. If a bound on ‖f(k)(.)‖ is given also, k≤m, then bounds on f(ζ) can be found for any ζ. A method of calculating the closest bounds is described, which is shown to be relevant to the problem of finding the interpolation formula whose error is bounded by the smallest possible multiple of ‖f(k)(.)‖, when ‖f(k)(.)‖ is unknown. This formula is identified and is called the optimal interpolation formula. The corresponding interpolating function is a spline of degree (k-1) with (m-k) knots, so it is very suitable for practical computation.

Keywords

Interpolation Problem Lagrange Interpolation Interpolation Formula Optimal Interpolation Optimal Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ahlberg, J.H., Nilson, E.N. and Walsh, J.L. (1967) “The theory of splines and their applications”, Academic Press, New York.zbMATHGoogle Scholar
  2. Gaffney, P.W. (1975) “Optimal Interpolation”, D. Phil thesis, University of Oxford.Google Scholar
  3. Hildebrand, F.B. (1956) “Introduction to numerical analysis”, McGraw-Hill Inc., New York.zbMATHGoogle Scholar
  4. Karlin, S. and Studden, W.J. (1966) “Tchebycheff systems: with applications in analysis and statistics”, Interscience Publishers, New York.zbMATHGoogle Scholar
  5. Micchelli, C.A., Rivlin, T.J. and Winograd, S. (1975) “The optimal recovery of smooth functions”, manuscript. IBM Research Laboratories, Yorktown Heights.zbMATHGoogle Scholar
  6. Schoenberg, I.J. (1964) “On interpolation by spline functions and its minimal properties”, from “On Approximation Theory”, eds. P.L. Butzer and J. Korevaar, Birkhaüser Verlag.Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  1. 1.Oxford University Computing LaboratoryOxford

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