Kernel-Based Methods: Milestones and New Trends

Kernel methods, with origins in the pioneering work of Mercer (1909), Bochner (1933), and Aronszajn (1950), have become central tools in modern mathematics and machine learning. This book explores their deep connections with approximation theory, highlighting both classical results and cutting-edge developments. Through clear explanations and illustrative examples, it guides readers from foundational concepts to contemporary applications, including computational methods and real-world problem solving. By bridging theory and practice, the text not only provides a rigorous understanding of kernels but also inspires further exploration and research. Suitable for students, researchers, and practitioners, it invites readers to engage with ongoing advances in this dynamic field and to contribute to its future growth.

Stefano De Marchi is Full Professor of Numerical Analysis at the University of Padova. He has authored more than 150 publications and is the managing editor of the open-access journal Dolomites Research Notes on Approximation. He is the founder of the Italian Network on Approximation and one of the discoverers of the so-called Padua points.

Francesco Marchetti is Assistant Professor of Numerical Analysis at the University of Padova. His research lies at the intersection of approximation theory and machine learning, focusing on kernel methods and polynomial and RBF interpolation, with applications to medical imaging and space weather forecasting, within national and international research projects.

Emma Perracchione is Associate Professor at Politecnico di Torino. Her research focuses on approximation theory and its applications to solar physics, such as astronomical imaging. She is currently Principal Investigator of the GOSSIP project (Greedy Optimal Sampling for Solar Inverse Problems), funded by the Italian Ministry of Universities and Research.