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On reproducing kernels, and analysis of measures. (English) Zbl 07116991

Summary: 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 ItĂ´ calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.

MSC:

47L60 Algebras of unbounded operators; partial algebras of operators
46N30 Applications of functional analysis in probability theory and statistics
46N50 Applications of functional analysis in quantum physics
42C15 General harmonic expansions, frames
65R10 Numerical methods for integral transforms
31C20 Discrete potential theory
62D05 Sampling theory, sample surveys
94A20 Sampling theory in information and communication theory
39A12 Discrete version of topics in analysis
46N20 Applications of functional analysis to differential and integral equations
22E70 Applications of Lie groups to the sciences; explicit representations
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
58J65 Diffusion processes and stochastic analysis on manifolds
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