Using the definition to find Fourier transforms

Introduction

If you want to use Fourier analysis, you must develop basic skills for finding Fourier transforms of functions on ℝ. In principle, you can always use the defining integral from the analysis equation to obtain F when f is given. In practice, you will quickly discover that it is not so easy to find a transform such as

by using the techniques of elementary integral calculus, see Ex. 1.1.

In this chapter, we will present a calculus (i.e., a computational process) for finding Fourier transforms of commonly used functions on ℝ. You will memorize a few Fourier transform pairs f, F and learn certain rules for modifying or combining known pairs to obtain new ones. It is analogous to memorizing that (xn)′ = nxn-1, (sin x)′ = cos x, (ex)′ = ex, … and then using the addition rule, product rule, quotient rule, chain rule, … to find derivatives. You will need to spend a bit of time mastering the details, so do not despair when you see the multiplicity of drill exercises!

Once you learn to find Fourier transforms, you can immediately use Fourier's analysis and synthesis equations, Parseval's identity, and the Poisson relations to evaluate integrals and sums that cannot be found by more elementary methods. You will also need these skills when you study various applications of Fourier analysis in the second part of the course.

The box function

The box function

and the cardinal sine or sinc function

are two of the most commonly used functions in Fourier analysis. We often simplify the definition of such functions …