Gateways Into Electronics

2

GENERALIZED FUNCTIONS AND INTEGRAL TRANSFORMS

2-1 INTRODUCTION

In Chapter 1 we assumed that it was legitimate to obtain the step response of a phase lead by using a limiting procedure; we will now put this assumption on a firm footing by introducing generalized functions. Despite their imposing name, generalized functions have surprisingly simple properties, and we will see that they offer a freedom of action that ordinary functions cannot match. We will also complete our program of obtaining a multiplicative operator for arbitrary inputs to a linear system by introducing Laplace transforms, whose natural setting is precisely among generalized functions, and we will give the frequency content of waveforms a formal basis by introducing Fourier transforms. Our examples will be straightforward, but our approach will be abstract; the payoff will be a powerful set of tools, applicable far beyond the context in which they will be developed.

We will be interested primarily in concepts and mechanics of use; this is why we will prove many minor theorems and accept a few difficult theorems: simple proofs will prepare us to understand what we are accepting. The difficult theorems generally involve the integrability of functions and the legitimacy of exchanging the order of limits; they will thus be intuitively reasonable on the one hand, and verifiable in specific cases on the other.

2-2 MATHEMATICAL BACKGROUND

The results we will obtain in this chapter are quite general, and are best stated ...