Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations

TL;DR

This paper develops a complex-valued version of the representer theorem in RKHSs, connecting superoscillations, Hardy and Fock spaces, and kernel methods in machine learning.

Contribution

It introduces a complex-reproducing kernel Hilbert space framework and links superoscillations with kernel methods, extending the theoretical understanding of these phenomena.

Findings

Recovered superoscillations in Fock space

Identified Gaussian RBF kernel solutions in RKHS

Connected superoscillations with Hardy space Blaschke products

Abstract

We introduce a complex-valued counterpart of the representer theorem in machine learning. We study several learning and minimization problems in reproducing kernel Hilbert spaces (RKHSs), with the aim of identifying appropriate input-output data sets that allow specific functions to appear as solutions of regression-type minimization problems. In particular, we recover superoscillations in the Fock space, the Gaussian radial basis function (RBF) kernel in the corresponding RKHS, and finite Blaschke products in the Hardy space setting. We then extend the notion of superoscillations through suitable generalizations of the Fock space and investigate the associated learning problems. This is a seminal work relating superoscillations and machine learning kernel methods via the representer theorem.