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Nonuniform sampling, reproducing kernels, and the associated Hilbert spaces. (English) Zbl 1393.47047

Summary: In a general context of positive definite kernels \(k\), we develop tools and algorithms for sampling in reproducing kernel Hilbert space \(\mathcal H\) (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction of an “entire” (or global) signal, a function \(f\) from \(\mathcal H\), via generalized interpolation of \(f\) from partial information obtained from carefully chosen distributions of sample points. We give necessary and sufficient conditions for configurations of point-masses \(\delta x\) of sample-points \(x\) to have finite norm relative to the particular RKHS \(\mathcal H\) considered. When this is the case, and the kernel \(k\) is given, we obtain an induced positive definite kernel \(\langle\delta_x,\delta_y\rangle_{\mathcal H}\). We perform a comparison, and we study when this induced positive definite kernel has \(l^2\) rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, \(l^2\) on one side, and the RKHS \(\mathcal H\) on the other. A number of applications are given, including to infinite network systems, to graph Laplacians, to resistance metrics, and to sampling of Gaussian fields.

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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