Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 1

Sampling with positive definite kernels and an associated dichotomy

Pages: 125 – 154



Palle Jorgensen (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

James Tian (Mathematical Reviews, Ann Arbor, Michigan, U.S.A.)


We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the property that there are countable discrete sample-subsets $S$; i.e., proper subsets $S$ having the property that every function in $\mathscr{H} (K)$ admits an $S$-sample representation. We give a characterizations of kernels which admit such non-trivial countable discrete sample-sets. A number of applications and concrete kernels are given in the second half of the paper.

Published 22 May 2020