n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

TL;DR

This paper proves the existence of optimal n-best kernel approximations in a broad class of reproducing kernel Hilbert spaces of holomorphic functions, extending classical approximation results and enabling applications in signal processing and numerical analysis.

Contribution

It generalizes the n-best approximation existence results to a wide class of RKHS of holomorphic functions and stochastic processes, using the maximum modulus principle.

Findings

Established existence of n-best kernel approximations in broad RKHS.

Extended classical rational approximation results to new function spaces.

Applied findings to signal processing and numerical solutions of equations.

Abstract

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of $n$-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of $n$-best rational approximation for the Hardy space and a recent result of $n$-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.