Math2111: Chapter 1: Fourier series. Section 1: Background information

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We start the lectures with \rhd Fourier series which go back to \rhd Joseph Fourier.

Fourier discovered what we now call Fourier series when he studied the heat equation

\displaystyle \frac{\partial u}{\partial t} - \alpha \left(\frac{\partial^2 u}{\partial x^2} +  \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)=0

We will study solutions to this problem using Fourier series later in this course, see also \rhd 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:

as well as many other applications. It also marks the beginning of \rhd harmonic analysis, an important area in mathematics.

What is the aim?

Let a periodic function f: [-\pi, \pi] \to \mathbb{R} be given. We want to represent the function as an infinite series of the basic periodic functions

\displaystyle \cos(kx), \sin(kx), \quad k = 0, 1, 2, \ldots.

More precisely, we are looking for a formula of the form

\displaystyle f(x) = \frac{a_0}{2} + \sum_{k=1}^\infty [a_k \cos (kx) + b_k \sin (kx)].

We call a_0, a_1, \ldots and b_1, b_2, \ldots the Fourier coefficients of f.

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 a_0, a_1, a_2, \ldots and b_1, b_2, b_3, \ldots for some given function f?
  • 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.

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