MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Christian Rieger and Barbara Zwicknagl


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.

MSC Codes:
41A05, 41A25, 41A63, 65D10, 68T05
Gaussians, inverse Multiquadrics, smoothing, approximation, radial basis functions, convergence orders

Related publications

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