Fourier discovered what we now call Fourier series when he studied the heat equation
We will study solutions to this problem using Fourier series later in this course, see also solution to heat equation.
Since then the ideas behind Fourier series have been generalized in many ways:
as well as others. These methods have found widespread applications, for instance:
- solving partial differential equations
- in number theory for the proof of the prime number theorem
- numerical analysis
- signal processing
- image compression ( jpeg)
as well as many other applications. It also marks the beginning of harmonic analysis, an important area in mathematics.
What is the aim?
Let a periodic function be given. We want to represent the function as an infinite series of the basic periodic functions
More precisely, we are looking for a formula of the form
We call and the Fourier coefficients of
A java applet illustrating such a representation can be found at http://www.falstad.com/fourier/.
The fundamental questions arising are:
- How can we calculate and for some given function ?
- Which functions can be represented in this way?
We give an answer to the questions above in the next few lectures. Before we do so we need to recall a few concepts from linear algebra and calculus, which we do in the next sections.