3. Fourier Theory

Fourier theory plays an important role in applied analysis. In this chapter we give an overview of the most important aspects needed in this book. First, we introduce an inner product and (orthonormal) basis functions in Section 3.1. Here we also define Fourier series, consider their convergence, and give Parseval's identity. We give both the complex and the trigonometric representations. Finally, the integral analogue of a series is introduced and exemplified. Next, in Section 3.3, the discrete form of the Fourier transform is considered, derived from the continuous version. Again, convergence and Parseval's identity are studied. Also, important phenomena, such as aliasing that show the restrictions of the discrete Fourier transform are treated. One very important application of this Fourier transform is in analysing linear equations with periodic boundary values. Despite the limitations of this problem class, it turns out that many physically meaningful concepts, such as stability, dissipation, and dispersion, can be studied quite fruitfully for the transformed equation, both in the continuous and in the discrete cases. In Section 3.4, therefore, the use of these transformations is demonstrated, leading to the important concept of the dispersion relation.