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Swapping edges of arbitrary triangulations to achieve the optimal order of approximation. (English) Zbl 0889.41014

Author’s summary: “In the representation of scattered data by smooth \(pp\) (:= piecewise polynomial) functions, perhaps the most important problem is to find an optimal triangulation of the given sample sites (called vertices). Of course, the notion of optimality depends on the desirable properties in the approximation or modeling problems. In this paper, we are concerned with optimal approximation order with respect to the given order \(r\) of smoothness and degree \(k\) of the polynomial pieces of the smooth \(pp\) functions. We will only consider \(C1\) \(pp\) approximation with \(r=1\) and \(k=4\). The main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites, such that these sample sites are the only vertices of the triangulation, and that for any discrete data given at these sample sites, there is a \(C1\) piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation”.

MSC:

41A25 Rate of convergence, degree of approximation
41A63 Multidimensional problems
41A05 Interpolation in approximation theory
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