1 Introduction
2 Fourier series
 2.1 Fourier series in terms of trigonometric functions
 2.2 Fourier series in terms of basis functions e^inπx∕L
 2.3 Numerical computation of Fourier expansion coefficient
3 Discrete Fourier transformation (DFT)
 3.1 Definition
 3.2 Periodic property of DFT
 3.3 Frequency resolution and bandwidth
4 Inverse transform
 4.1 Reconstruct the original function using DFT
 4.2 Evaluate the reconstructed function at discrete points
 4.3 Inverse Discrete Fourier transformation
 4.4 Proof of the inverse DFT
5 About using the FFT library
6 Discrete sine transform and cosine transform
 6.1 Traditional defintion of discrete Sine Transform (DST)
 6.2 Define DST via DFT
 6.3 Reconstruct original function using DST
 6.4 Discrete Cosine transform
7 Misc content
 7.1 Nonlinear process
 7.2 Aliasing errors
 7.3 Relation between Fourier series coefficients and DFTs
A Efficient method of computing DFT: Fast Fourier Transformation (FFT) algorithm (not finished)
B Fourier series→Fourier transformation
 B.1 From discrete spectrum to continuous spectrum
 B.2 Fourier transformation
 B.3 Numerical computation of Fourier transformation
 B.4 Sampling theorem
C 2D Fourier series
 C.1 For 2D real-valued functions
 C.2 2D real-valued Fourier series derived directly from real-valued trigonometric functions expansion– to be deleted, because there is an easier way to do this, as is given by the above section
D Details on FFT codes provided by the Numerical recipes book[2]
 D.1 Computing Fourier integrals using FFT (not finished, to be deleted)
References