A bridge between the rigorous foundations and the many rich applications of Fourier Analysis.
The first of its kind, this focused textbook serves as a self-contained resource for teaching from scratch the fundamental mathematics of Fourier analysis and illustrating some of its most current, interesting applications, including medical imaging and radar processing. Developed by the author from extensive classroom teaching experience, it provides a breadth of theory that allows students to appreciate the utility of the subject, but at as accessible a depth as possible. With myriad applications included, this book can be adapted to a one or two semester course in Fourier Analysis or serve as the basis for independent study.
Applied Fourier Analysis was created to bridge the gap between mathematics, engineering, physics, and computer science and other sciences.
This course will allow students from Mathematics, Physics, Engineering and elsewhere to gain some mathematical rigor in Fourier Analysis, as well as understand some of the many rich applications of Fourier Analysis.
The book is built around three fundamental chapters: Fourier Series, The Discrete Fourier Transform, and The Continuous Fourier Transform. The interplay between these three approaches is emphasized throughout. The basic concepts of Convolution, The Fourier Isometry, Differentiation and Decay rates are compared between chapters. The Uncertainty Principle and the other basic Fourier Identities are intertwined to give a great base for applications.
Applications include: Sampling and Interpolation, Medical Imaging, Image Processing, Radar Imaging, PDE's, and Digital Communications. There are numerous videos throughout the book, available through both hard and electronic copies, which illustrate all of the theory and concepts.
Applied Fourier Analysis assumes no prior knowledge of analysis from its readers, and begins by making the transition from linear algebra to functional analysis. It goes on to cover basic Fourier series and Fourier transforms before delving into applications in sampling and interpolation theory, digital communications, radar processing, medical imaging, and heat and wave equations.
For all applications, ample practice exercises are given throughout, with collections of more in-depth problems built up into exploratory chapter projects. Illuminating videos are available on Springer.com and Link.Springer.com that present animated visualizations of several concepts.
The content of the book itself is limited to what students will need to begin to understand the wide variety within these various academic fields. The approach avoids spending undue time studying proofs or building toward more abstract concepts.
The book is perhaps best suited for courses aimed at upper division undergraduates and early graduates in mathematics, electrical engineering, mechanical engineering, computer science, physics, and other natural sciences. It can also be a highly valuable resource for introducing a broad range of students to Fourier analysis, or for self-study.
Choose your favorite form of media for this book.
Both give access to embedded videos