Modern Perspectives in Approximation Theory

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.