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Positive thin plate splines. (English) Zbl 0595.41012

Summary: We consider the problem of interpolating positive data at scattered data points of the plane \(R^ 2\). To solve the problem, we introduce the positive thin plate spline, i.e. the solution to \[ Minimize\int_{R^ 2}\{\frac{\partial^ 2u}{\partial x^ 2}\}^ 2+2\{\frac{\partial^ 2u}{\partial x\partial \quad y}\}^ 2+\{\frac{\partial^ 2u}{\partial y^ 2}\}^ 2, \] \(u\in D^{-2}L^ 2(R^ 2)\); \(u(t_ j)=z_ j\), \(j=1,...,n\); u(t)\(\geq 0\), \(t\in K\) where \((t_ j,z_ j)\) are the data, K is a convex compact subset of \(R^ 2\). We give existence, uniqueness, characterisation and convergence results. We also present a dual algorithm to compute this spline and show numerical experience with the method.

MSC:

41A15 Spline approximation
41A05 Interpolation in approximation theory
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