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MiS Preprint
151/2006

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

Christian Rieger and Barbara Zwicknagl

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.

Received:
Dec 14, 2006
Published:
Dec 14, 2006
MSC Codes:
41A05, 41A25, 41A63, 65D10, 68T05
Keywords:
Gaussians, inverse Multiquadrics, smoothing, approximation, radial basis functions, convergence orders

Related publications

inJournal
2010 Journal Open Access
Christian Rieger and Barbara Zwicknagl

Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning

In: Advances in computational mathematics, 32 (2010) 1, pp. 103-129