Reproducing Kernel Hilbert Space vs. Frame Estimates

Palle Jorgensen,Myung-Sin Song

doi:10.3390/math3030615

Palle Jorgensen, Myung-Sin Song

Open Access

https://doi.org/10.3390/math3030615

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Abstract

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.

Highlights

A reproducing kernel Hilbert space (RKHS) is a Hilbert space H of functions on a set, say Ω, with the property that f (t) is continuous in f with respect to the norm in H
It is called reproducing because it reproduces the function values for f in H. Reproducing kernels and their RKHSs arise as inverses of elliptic PDOs, as covariance kernels of stochastic processes, in the study of integral equations, in statistical learning theory, empirical risk minimization, as potential
Mathematics 2015, 3 kernels, and as kernels reproducing classes of analytic functions, and in the study of fractals, to mention only some of the current applications. They were first introduced in the beginning of the 20ties century by Stanisaw Zaremba and James Mercer, Gábor Szegö, Stefan Bergman, and Salomon Bochner

Summary

Introduction

A reproducing kernel Hilbert space (RKHS) is a Hilbert space H of functions on a set, say Ω, with the property that f (t) is continuous in f with respect to the norm in H. It is called reproducing because it reproduces the function values for f in H. Mathematics 2015, 3 kernels, and as kernels reproducing classes of analytic functions, and in the study of fractals, to mention only some of the current applications. They were first introduced in the beginning of the 20ties century by Stanisaw Zaremba and James Mercer, Gábor Szegö, Stefan Bergman, and Salomon Bochner. Our aim in the present paper is to point out an intriguing use of reproducing kernels in the study of frames in Hilbert space

An Explicit Isomorphism

Frames and Gaussian Processes