Fourier Transforms

Abstract

Later in this chapter we define the Fourier Transform. There are two ways of approaching the subject of Fourier Transforms, both ways are open to us! One way is to carry on directly from Chapter 4 and define Fourier Transforms in terms of the mathematics of linear spaces by carefully increasing the period of the function f(x). This would lead to the Fourier series we defined in Chapter 4 becoming, in the limit of infinite period, an integral. This integral leads directly to the Fourier Transform. On the other hand, the Fourier Transform can be straightforwardly defined as an example of an integral transform and its properties compared and in many cases contrasted with those of the Laplace Transform. It is this second approach that is favoured here, with the first more pure mathematical approach outlined towards the end of Section 6.2. This choice is arbitrary, but it is felt that the more “hands on” approach should dominate here. Having said this, texts that concentrate on computational aspects such as the FFT (Fast Fourier Transform), on time series analysis and on other branches of applied statistics sometimes do prefer the more pure approach in order to emphasise precision.