Improved Exponential Convergence Rates by Oversampling Near the Boundary
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Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.
KeywordsSampling inequalities Convergence orders Boundary effect Smooth kernels Gaussian Inverse multiquadrics Chebyshev nodes
Mathematics Subject Classification41A05 41A25 41A63 65D05
We thank R. Schaback, M. Griebel, and T. Hangelbroek for helpful discussions, and the referees for their careful reading and helpful comments. Some preliminary results are contained in C.R.’s PhD thesis defended at the University of Göttingen (see ). C.R. acknowledges partial support by the DFG through the Collaborative Research Centers (SFB) 611 and 1060, and the Hausdorff Center for Mathematics. B.Z. warmly thanks the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where part of this research was carried out. Her research was partly funded by a postdoctoral fellowship of the National Science Foundation under Grant No. DMS-0905778.
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