Based on the C.I.M.E. Summer School held in Cetraro (Italy) in July 2025, this volume presents a unified perspective on fundamental questions of stability, efficiency, and spectral structure in modern approximation theory. It collects recent developments in kernel-based approximation, harmonic analysis, graph signal processing, and sampling theory, with particular attention to the representation and reconstruction of functions and signals on irregular domains.

The book consists of three chapters. Quasi-interpolation and Radial Basis Functions by Martin D. Buhmann; Harmonic Analysis on Graphs and Networks by Wolfgang Erb; and Exponential Bases and Sampling Theory by Götz Pfander, which also share principles such as localization and spectral analysis by illustrating the interplay between classical theory and emerging applications.

The volume is intended for graduate students and researchers in approximation theory, harmonic analysis, numerical analysis, signal processing, and related areas.

Martin Buhmann is a full professor of numerical analysis at the Justus Liebig University Giessen, Germany. His research focuses on multivariate approximation theory, kernel methods, and radial basis functions, with contributions to nonlinear optimization and numerical differential equations. He is the author of Radial Basis Functions: Theory and Implementations and the co-author of Quasi-Interpolation (Cambridge University Press).

Stefano De Marchi is a full professor of numerical analysis at the University of Padova. His research spans approximation theory, kernel methods, multivariate interpolation, and topological data analysis. A co-discoverer of the Padua points, he has authored over 160 publications and delivered more than 120 invited lectures. He chairs the UMI Thematic Group on Approximation Theory and Applications and is the Managing Editor of Dolomites Research Notes on Approximation.

Gianluca Vinti is a full professor of mathematical analysis at the University of Perugia. His research focuses on approximation theory, signal and image processing, and mathematical models for biomedical and engineering applications. He has authored over 180 scientific publications and a renowned monograph, holds patents in biomedical imaging, and has delivered more than 150 invited lectures at universities and research centers worldwide.

Wolfgang Erb is an associate professor of numerical analysis at the University of Padua, Italy. His research focuses on computational harmonic analysis, graph learning and signal processing, multivariate approximation, inverse problems, and reconstruction algorithms for biomedical imaging, particularly magnetic particle imaging. He earned his Ph.D. from the Technical University of Munich and has held postdoctoral positions in Germany and the USA.

Götz E. Pfander is a full professor of mathematics and the Chair of Scientific Computing at the Katholische Universität Eichstätt–Ingolstadt, Germany. His research covers harmonic and time-frequency analysis, sampling theory, frame theory, compressed sensing, and data science. He is the Editor-in-Chief of Sampling Theory, Signal Processing, and Data Analysis, the Chair of the SampTA Steering Committee, and has led numerous research projects funded by the German Research Foundation (DFG).