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Approximate interpolation with applications to selecting smoothing parameters. (English) Zbl 1081.65016

An approximation problem can be shortly stated as follows: for a finite set \(X\) of points situated in a bounded set \(\Omega\) and a corresponding data values of an unknown function \(f \in C(\Omega)\), a function \(s_{f,X} \in C(\Omega)\) to produce a good approximation is required. The authors investigate how a small error at the data set influences the global error on \(\Omega\). An estimation of the global error is given and the main application refers to the spline smoothing. It is shown that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. An application in stabilizing the interpolation process by splines and positive definite kernels is presented and the final section gives some eloquent numerical examples.

MSC:

65D05 Numerical interpolation
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
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