Reproducing kernel Hilbert spaces, polynomials, and the classical moment problem

Dette, Holger and Zhigljavsky, Anatoly A. ORCID: https://orcid.org/0000-0003-0630-8279 2021. Reproducing kernel Hilbert spaces, polynomials, and the classical moment problem. SIAM/ASA Journal on Uncertainty Quantification 9 (4) , pp. 1589-1614. 10.1137/21M1394965

Preview

PDF - Accepted Post-Print Version Download (627kB) | Preview

Abstract

We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to 0 implies that the regression function cannot be an element of the reproducing kernel Hilbert space. This question is then related to the determinacy of the Hamburger moment problem for the spectral measure corresponding to the kernel. In the literature it was observed that a nonvanishing constant function does not belong to the reproducing kernel Hilbert space associated with the Gaussian kernel. Our results provide a unifying view of this phenomenon and show that the mentioned result can be extended for arbitrary polynomials and a broad class of translation-invariant kernels.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	2166-2525
Date of First Compliant Deposit:	29 November 2021
Date of Acceptance:	3 November 2021
Last Modified:	27 Nov 2024 18:15
URI:	https://orca.cardiff.ac.uk/id/eprint/145811

Actions (repository staff only)

Edit Item