Pointwise optimal multivariate spline method for recovery of twice differentiable functions on a simplex
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We obtain the spline recovery method on a d-dimensional simplex T that uses as information values and gradients of a function f at the vertices of T and is optimal for recovery of f( w) at every point w of an admissible domain P containing T on the class W^2(P) of twice differentiable functions on P with uniformly bounded second order derivatives in any direction. If, in particular, every face of T (of any dimension) contains its circumcenter, we can take P=T. We also find the error function of the pointwise optimal method which turns out to be a function in W^2(P) with zero information. The error function is a piecewise quadratic C^1-function over a certain polyhedral partition and can be considered as a multivariate analogue of the classical Euler spline ϕ_2. The pointwise optimal method is a continuous spline of degree two (with some pieces of degree one) over the same partition.
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