[next] [prev] [prev-tail] [tail] [up]

Chapter 5
Fourier Series Analysis

x−π−1NNN π1=== 1131

Figure 5.1: Partial sums of the Fourier series of a square wave, with \(N=1\), \(3\) and \(11\) non-zero terms. The approximation improves everywhere except near the jump, where the overshoot persists: it narrows as terms are added but does not shrink, settling at about \(9\%\) of the jump. This is the Gibbs phenomenon, and it is why convergence of a Fourier series is stated in the mean-square sense of Theorem 4.5.1 rather than uniformly.

Another way of representing functions in terms of series is by representing as series of sines and cosines which we call Fourier series. Applications of Fourier series range from solutions of PDE’s, time series analysis, engineering to many other applications.

5.1 The Series and Its Coefficients
5.2 Periodicity and Periodic Extension
5.3 Worked Examples
5.4 The Dirichlet Kernel
5.5 Convergence of Fourier Series
5.6 Symmetry and Half-Range Expansions
5.7 Parseval’s Identity
5.8 The Complex Form
5.9 From Fourier Series to the Fourier Transform
5.10 Practice Problems

[next] [prev] [prev-tail] [front] [up]