This expository article presents the approach to statistical machine learning based on reproducing kernel Hilbert spaces. The basic framework is introduced for scalar-valued learning and then extended to operator learning. Finally, learning dynamical systems is formulated as a suitable operator learning problem, leveraging Koopman operator theory. The manuscript collects the supporting material for the corresponding course taught at the CIME school "Machine Learning: From Data to Mathematical Understanding" in Cetraro.
@article{arxiv.2509.18071,
title = {Learning functions, operators and dynamical systems with kernels},
author = {Lorenzo Rosasco},
journal= {arXiv preprint arXiv:2509.18071},
year = {2025}
}