Fourier Transforms

CHAPTER 6 GENERALIZED FUNCTIONS

The Fourier transform developed in the previous chapter applies to functions which are absolutely or square integrable on the real line (members of or ). These classes are large and encompass many functions of importance in engineering and physics, but there are some surprising omissions—sine, cosine, the step and signum functions, and powers of x, to name a few. These are brought into the picture by the introduction of generalized functions, the subject of this chapter.

The best-known generalized function is the delta function δ(x), which models impulsive phenomena like sudden shocks and point charge or mass distributions. The chapter begins with some physical situations that naturally lead to the introduction of the delta function. Some operational rules for manipulating delta functions and using them for practical calculations are then developed. This leads into a broader discussion of generalized function theory. It is shown that all the ordinary functions we have worked with so far, and more, are in fact generalized functions, as well as objects like the delta function with behaviors that defy the traditional definition of “function.” All generalized functions possess derivatives of all orders—continuity is no longer a restriction. All generalized ...