Reproducing kernel Hilbert spaces, polynomials and the classical moment problems

TL;DR

This paper investigates the relationship between polynomials, reproducing kernel Hilbert spaces (RKHS), and classical moment problems, establishing conditions under which polynomials do not belong to certain RKHS associated with translation-invariant kernels.

Contribution

It provides a unifying framework linking the non-inclusion of polynomials in RKHS to the determinacy of the Hamburger moment problem for spectral measures.

Findings

Polynomials are not in the RKHS of certain translation-invariant kernels.

The non-inclusion relates to the determinacy of the Hamburger moment problem.

Extends known results about constant functions in Gaussian RKHS to broader classes.

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 non-vanishing constant function does not belong to the reproducing kernel Hilbert space associated with the Gaussian kernel (see Corollary 4.44 in Steinwart and Christmann, 2008). 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.