ABSTRACT
This chapter discusses the Fourier transform, which is one of the most important tools in pure and applied mathematics. It describes its principal properties, including the concept of convolution, using FAS to provide many illustrations. Several applications to mathematical physics and signal processing will be described. The chapter examines some important topics in communication theory, Poisson summation and sampling theory. Although it is possible to motivate the definition of the Fourier transform by way of Fourier series, it is logically simpler to just begin with the definition of the Fourier transform and then show its properties and applications. The chapter explores some inversion theorems for Fourier transforms. It covers some important properties of Fourier transforms, for example, how they behave with respect to shifting, scaling, and differentiation. Besides Fourier transforms, there are two other transforms commonly used in Fourier analysis, the Fourier sine and Fourier cosine transforms.
