Fourier Series

In this chapter we will consider approximating a function by a linear combination of basis functions, which are simple functions that can be generated in a laboratory. Joseph Fourier (1768–1830) developed the mathematical theory of heat conduction using a set of trigonometric (sine and cosine) series of the form we now call Fourier series (Fourier, J.B.J., 1955 (A. Freeman, translation)). He established that an arbitrary mathematical function can be represented by its Fourier series. This idea was new and startling and met with vigorous opposition from some of the leading mathematicians at the time, see Hawking (2005). Fourier series and the Fourier transform are basics to mathematics and science, especially to the theory of communications. For example, a phoneme in a speech signal is smooth and wavy. A linear combination of a few sinusoidal functions would approximate a segment of speech within some error tolerance. Suppose we like to build a structure that allows us to climb from the first floor to the second floor of a building. We can have a staircase approximating a ramp function using a linear combination of pulse functions. The amplitudes and the width of the pulses can be determined based on the error between the ramp and the staircase. Apart from the staircase problem, this type of analysis is important in electrical engineering, for example, when converting an analog signal to a discrete signal.