New technological innovations and advances in research in areas such as spectroscopy, computer tomography, signal processing, and data analysis require a deep understanding of function approximation using Fourier methods. To address this growing need, this monograph combines mathematical theory and numerical algorithms to offer a unified and self-contained presentation of Fourier analysis. The first four chapters of the text serve as an introduction to classical Fourier analysis in the univariate and multivariate cases, including the discrete Fourier transforms, providing the necessary background for all further chapters. Next, chapters explore the construction and analysis of corresponding fast algorithms in the one- and multidimensional cases. The well-known fast Fourier transforms (FFTs) are discussed, as well as recent results on the construction of the nonequispaced FFTs, high-dimensional FFTs on special lattices, and sparseFFTs. An additional chapter is devoted to discrete trigonometric transforms and Chebyshev expansions. The final two chapters consider various applications of numerical Fourier methods for improved function approximation, including Prony methods for the recovery of structured functions. This new edition has been revised and updated throughout, featuring new material on a new Fourier approach to the ANOVA decomposition of high-dimensional trigonometric polynomials; new research results on the approximation errors of the nonequispaced fast Fourier transform based on special window functions; and the recently developed ESPIRA algorithm for recovery of exponential sums, among others. Numerical Fourier Analysis will be of interest to graduate students and researchers in applied mathematics, physics, computer science, engineering, and other areas where Fourier methods play an important role in applications.

Autorentext

Gerlind Plonka received the Ph.D. degree in mathematics and the Habilitation degree from the University of Rostock in 1993 and 1996, respectively. She was an Associate Professor of applied analysis at the University of Duisburg-Essen, Germany, from January 1998 to June 2010. Since July 2010, she works as a Full Professor of applied mathematics at the University of Göttingen, Germany. Her current research interests include numerical methods of Fourier analysis, wavelet theory, and inverse problems with applications to signal and image processing. Daniel Potts received his Ph.D. degree in mathematics from the University of Rostock in 1998.He held a research position at the University of Lübeck from 1996 to 2005, where he obtained his Habilitation degree in 2004. Since 2005 he works as a Full Professor at the TU Chemnitz. His research focuses on applied analysis, in particular computational harmonic analysis, and applications in scientific computing. Gabriele Steidl received the Ph.D. degree in mathematics and the Habilitation degree from the University of Rostock in 1988 and 1991, respectively. She had positions as Associated Professor for mathematics at the TU Darmstadt and as Full Professor at the University of Mannheim and the TU Kaiserslautern, where she also worked as consultant of the Fraunhofer ITWM Kaiserslautern. Since 2020 she is Full Professor at the TU Berlin. In 2022 she became a SIAM Fellow. Her research interests include harmonic analysis, optimization, inverse problems and machine learning with applications in image and signal processing. Manfred Tasche received the Ph.D. degree in mathematics and the Habilitation degree from the University of Rostock in 1966 and 1976, respectively. He was Associate Professor and later Full Professor for Analysis and Numerical Mathematics at the University of Rostock from 1978 to 1993. Until 2008, he worked as substitute Professor and Assistant Professor at the University of Lübeck and the University of Rostock. Since 2008 he is retired. His research focuses on numerical analysis and approximation, Fourier analysis and applications in signal processing.