Reproducing kernel Hilbert spaces, polynomials and the classical moment problems
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Date
2021
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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.
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Keywords
reproducing kernel Hilbert spaces, continuous time regression model, best linear estimation, classical moment problem,