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Abstract
In this paper, we investigate the difference of Shepard's generalized operators S σ from the approximated set of data for various weight functions σ. Bounds are given for the sizes of the ‘bumps’ shown on the graph of S σ for σ(d)=1/d in dimension N=1, and the best weight function σ for practical use is proposed.
Acknowledgements
The authors thank Balázs Szalkai for a lot of computer help and the referees for their valuable remarks.
Notes
In practice, these data are obtained by measuring and not by using a formula.
Which can be embedded in some higher dimensional space.
The assumption 0<τ<1 is not a restriction in fact, since the limit we are discussing in this paper is the same for any fixed point
. This is why we may restrict ourselves to the interval (P
0, P
1).