Numerische Mathematik

, Volume 101, Issue 4, pp 729–748

# Approximate Interpolation with Applications to Selecting Smoothing Parameters

• Holger Wendland
• 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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