In the previous chapter we encountered the Fourier series representation of a periodic function in a fixed interval as a superposition of sinusoidal functions. It is often desirable, however, to obtain such a representation for functions that are defined over an infinite interval and have no particular periodicity. Such a representation is called a Fourier transform and is one of a class of representations called integral transforms.
We begin by considering Fourier transforms as a generalization of Fourier series. We then go on to discuss the properties of the Fourier transform and its applications. In the second part of the chapter we present an analogous discussion of the closely related Laplace transform.
5.1 Fourier ...