A sharp upper bound for sampling numbers in L_2

Matthieu Dolbeault (Sorbonne U, Paris)

May 10. 2022, 13:50 — 14:40

For a class F of complex-valued functions on a set D, we define its sampling numbers g_n

as the minimal worst-case error on F, measured in L_2, that can be achieved with an optimal algorithm

based on n function evaluations. We prove that there is a universal constant > 0 such that

the sampling numbers of the unit ball F of every separable reproducing kernel Hilbert space are bounded by

the tails of the the Kolmogorov widths (or approximation numbers) of F in L_2.

We also obtain similar upper bounds for more general classes F, including compact subsets of

the space of continuous functions on a bounded domain D.

Moreover, we provide examples where these bounds are attained up to a constant.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Computational Uncertainty Quantification: Mathematical Foundations, Methodology & Data (Thematic Programme)
Clemens Heitzinger (TU Vienna)
Fabio Nobile (EPFL Lausanne)
Robert Scheichl (U Heidelberg)
Christoph Schwab (ETH Zürich)
Sara van de Geer (ETH Zürich)
Karen Willcox (U of Texas, Austin)