Reproducing kernel Hilbert spaces, polynomials and the classical moment problems

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dc.contributor.authorDette, Holger
dc.contributor.authorZhigljavsky, Anatoly
dc.date.accessioned2021-01-29T13:28:52Z
dc.date.available2021-01-29T13:28:52Z
dc.date.issued2021
dc.description.abstractWe 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.en
dc.identifier.urihttp://hdl.handle.net/2003/40015
dc.identifier.urihttp://dx.doi.org/10.17877/DE290R-21898
dc.language.isoende
dc.relation.ispartofseriesDiscussion Paper / SFB823;4/2021en
dc.subjectreproducing kernel Hilbert spacesen
dc.subjectcontinuous time regression modelen
dc.subjectbest linear estimationen
dc.subjectclassical moment problem,en
dc.subject.ddc310
dc.subject.ddc330
dc.subject.ddc620
dc.titleReproducing kernel Hilbert spaces, polynomials and the classical moment problemsen
dc.typeTextde
dc.type.publicationtypeworkingPaperde
dcterms.accessRightsopen access
eldorado.secondarypublicationfalsede

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