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Numerische Mathematik

, Volume 101, Issue 4, pp 729–748 | Cite as

Approximate Interpolation with Applications to Selecting Smoothing Parameters

  • Holger WendlandEmail author
  • Christian Rieger
Article

Abstract

In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function might be interpreted as the error function between an unknown function and a given approximant. We will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels.

Mathematics Subject Classification (2000)

65D10 65D07 41A25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

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