Abstract
Sampling inequalities give a precise formulation of the fact that a differentiable function cannot attain large values if its derivatives are bounded and if it is small on a sufficiently dense discrete set. Sampling inequalities can be applied to the difference of a function and its reconstruction in order to obtain (sometimes optimal) convergence orders for very general possibly regularized recovery processes. So far, there are only sampling inequalities for finitely smooth functions, which lead to algebraic convergence orders. In this paper, the case of infinitely smooth functions is investigated, in order to derive error estimates with exponential convergence orders.
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Communicated by Joe Ward.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Rieger, C., Zwicknagl, B. Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. Adv Comput Math 32, 103 (2010). https://doi.org/10.1007/s10444-008-9089-0
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DOI: https://doi.org/10.1007/s10444-008-9089-0
Keywords
- Gaussians
- Inverse multiquadrics
- Smoothing
- Approximation
- Error bounds
- Radial basis functions
- Convergence orders