Multivariate splines

Hi,

In one dimension, a very simple and efficient B-spline-approximation of f (being in C^2) is given by "Schoenberg's variation diminishing spline approximant"

Vf= \sum_i f(t_i) B_i,

where each evaluation point t_i is in the support of B_i. This approximation is marred by an error of type

\sup|Vf-f| < Const \Delta^2 \sup|\nabla^2 f|,

with \Delta denoting the mesh-size and D the differential operator. I'm a novice within this field and my question is: Is there a multivariate counterpart to this approximant - that is, a local and multivariate spline approximation, obtainable from simple pointwise evaluations of f - having the same type of error?

Thanks in advance,

Jimmy



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