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.
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Wendland, H., Rieger, C. Approximate Interpolation with Applications to Selecting Smoothing Parameters. Numer. Math. 101, 729–748 (2005). https://doi.org/10.1007/s00211-005-0637-y
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DOI: https://doi.org/10.1007/s00211-005-0637-y