Sampling Inequalities for Infinitely Smooth Functions, with Applications to Interpolation and Machine Learning

Christian Rieger and Barbara Zwicknagl

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Submission date: 14. Dec. 2006 (revised version: April 2008)
Pages: 27
published in: Advances in computational mathematics, 32 (2010) 1, p. 103-129 
DOI number (of the published article): 10.1007/s10444-008-9089-0
Bibtex
MSC-Numbers: 41A05, 41A25, 41A63, 65D10, 68T05
Keywords and phrases: Gaussians, inverse Multiquadrics, smoothing, approximation, radial basis functions, convergence orders
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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.