On Reproducing Kernels, and Analysis of Measures

P. Jorgensen, F. Tian

2019, v.25, Issue 3

ABSTRACT

Starting with the correspondence between positive definite kernels
on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the
other, we turn to a detailed analysis of associated measures and Gaussian
processes. Point of departure: Every positive definite kernel is also
the covariance kernel of a Gaussian process.

Given a fixed sigma-finite measure $\mu$, we consider positive definite
kernels defined on the subset of the sigma algebra having finite $\mu$
measure. We show that then the corresponding Hilbert factorizations
consist of signed measures, finitely additive, but not automatically
sigma-additive. We give a necessary and sufficient condition for when
the measures in the RKHS, and the Hilbert factorizations, are sigma-additive.
Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast,
when $\mu$ is known to be atomic, our setting is shown to generalize
that of Shannon-interpolation. Our RKHS-approach further leads to
new insight into the associated Gaussian processes, their It\^{o}
calculus and diffusion. Examples include fractional Brownian motion,
and time-change processes.

Keywords: Hilbert space, reproducing kernel Hilbert space, harmonic analysis, Gaussian free fields, transforms, covariance

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