This book, "Fourier Analysis and Approximation," is a comprehensive treatise on Fourier analysis and its applications in approximation theory. Authored by Paul L. Butzer and Rolf J. Nessel, it is divided into five parts: Approximation by Singular Integrals, Fourier Transforms, Hilbert Transforms, Characterization of Certain Function Classes, and Saturation Theory. The first half focuses on Fourier series and Fourier integrals from a transform perspective, while the second half delves into convolution integrals and their applications in approximation theory. The book aims to provide a systematic treatment of Fourier analysis on the circle and the infinite line, emphasizing the underlying principles and culminating in saturation theory for convolution integrals. It is designed to serve as an introduction to these fields for advanced undergraduate students and researchers in mathematics and related sciences, with a strong emphasis on rigorous mathematical proofs and practical applications. The book includes approximately 550 problems and detailed references, making it a valuable resource for both learning and research.This book, "Fourier Analysis and Approximation," is a comprehensive treatise on Fourier analysis and its applications in approximation theory. Authored by Paul L. Butzer and Rolf J. Nessel, it is divided into five parts: Approximation by Singular Integrals, Fourier Transforms, Hilbert Transforms, Characterization of Certain Function Classes, and Saturation Theory. The first half focuses on Fourier series and Fourier integrals from a transform perspective, while the second half delves into convolution integrals and their applications in approximation theory. The book aims to provide a systematic treatment of Fourier analysis on the circle and the infinite line, emphasizing the underlying principles and culminating in saturation theory for convolution integrals. It is designed to serve as an introduction to these fields for advanced undergraduate students and researchers in mathematics and related sciences, with a strong emphasis on rigorous mathematical proofs and practical applications. The book includes approximately 550 problems and detailed references, making it a valuable resource for both learning and research.