Fourier’s Wonderful Idea – I

Breaking Complex Objects into Simple Pieces

“In a memorable session of the French Academy on the
21st of December 1807, the mathematician and engineer
Joseph Fourier announced a thesis which inaugurated a
new chapter in the history of mathematics. The claim of
Fourier appeared to the older members of the Academy,
including the great analyst Lagrange, entirely incredible.”

Introduction

Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

Joseph Fourier

Fourier, son of a tailor, was born just 250 years ago on 21 March 1768, and orphaned at the age of nine. During the 1798 campaign to colonize Egypt, Napoleon brought more than 150 scientists to study and catalogue Egyptian culture. Among them was Joseph Fourier.

After his return to France in 1801, Fourier began experiments on the diffusion of heat. He developed a partial differential equation — the heat equation — which describes how the temperature in a conductor changes over time. It is the classical example of a `parabolic equation’ and may be written for the case of a one-dimensional conductor, such as a metal bar, as

$\displaystyle \frac{\partial T}{\partial t} = \nu \frac{\partial^2 T}{\partial x^2}$

Its more general three-dimensional cousin describes a vast range of phenomena, from heat-flow in a cup of coffee to the temperature distribution within the Sun.

A Simple Solution

The temperature is a function of both space and time ${T = T(x,t)}$ . Fourier began by assuming that the independent variables in the temperature can be separated:

$\displaystyle T(x,t) = \xi(x)\tau(t)$

This works for sinusoidal space dependence, ${\xi(x)=\sin kx}$ . Then the time dependence is a decaying exponential ${\tau(t) = \tau_0\exp(-t/t_0)}$ . Substitution into the heat equation gives the time-scale for decay, ${t_0 = 1/\nu k^2}$ so that the solution becomes

$\displaystyle T(x,t) = \tau_0 \exp(- \nu k^2 t)\sin kx$

The sine-wave distribution decays exponentially to a constant value as heat is distributed equally throughout the bar.

Decaying-Sine-Wave

A More General Solution

Suppose now that the initial temperature varies in a non-sinusoidal manner. For example, it may change linearly from hot at one end to cold at the other. Fourier claimed that any temperature distribution could be expressed as an infinite sum of sine waves. This was what caused consternation at the Academy.

With certain technical but important qualifications, Fourier’s claim was justified and his insight led to dramatic developments in mathematics, both pure and applied. Fourier’s infinite sums could represent functions whose graphs had angles (discontinuous derivatives) and even functions with discontinuous jumps. This revolutionized our conception of a mathematical function.

Inversion & Duality

For each periodic function ${f(x)}$ , Fourier’s sum involves another function ${F(n)}$ which gives the weight to be assigned to each sine-wave or scale. There is a duality between the two functions. From ${f(x)}$ we can calculate ${F(n)}$ , and if we know ${F (n)}$ we can recover ${f(x)}$ . This process enables any signal to be decomposed into a spectrum of wavelengths.

A function and its transform are complementary: the function shows the time information whereas the information about frequencies is implicit but obscure. Since a sine wave extends over an infinite time, local properties of the signal, for example spikes in the input, become global features of the transform. Thus, a narrow spike in the input becomes a superposition of a large number of frequencies.

More on Fourier next week.

Sources

Lanczos, Cornelius, 1966: Discourse on Fourier Series, Oliver and Boyd. Republished by SIAM, 2016. ISBN: 978-1-61197-451-5. eISBN: 978-1-61197-452-2

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