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

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.