Chapter 6 - Fourier Analysis

Abstract

Any periodic function can be expressed as the sum of sine and cosine functions. For example, a musical instrument usually creates several different tones at the same time, which are linear combinations of sine and cosine functions with different frequencies. Outside mathematics, Fourier analysis is used in many areas. It contains all of the central ideas of electrical engineering. Crystallography, the telephone, the x-ray machine, and many other devices use Fourier's ideas. Following Fourier's original ideas, this chapter illustrates how to represent a periodic function as a series of sine and cosine functions. Fourier series are a useful tool for analyzing the frequency properties of a function. When two Fourier series are multiplied together, it is easier to use the complex exponential representation instead of the sine and cosine representation. The Fourier transform represents a function in the frequency domain. It is the continuous version of the Fourier series. Many results for the Fourier transform are parallel to those for Fourier series. Convolution is a useful tool in the study of relationships between functions. The convergence theorem for Fourier series confirms that a periodic function can be identified with its Fourier series.