= Fourier analysis
{c}
{wiki}
<Fourier analysis> describes a <function> by the <oscillations> that compose it. A graph records values at successive positions or times; a <frequency spectrum> records how much of each <frequency> is present, together with its <wave phase>. The same complicated shape can become a short list of simple components. For <periodic functions>, the components form a <Fourier series>; on the whole real line, the <Fourier transform> uses a continuous range of <frequencies>, and the <Fourier inversion theorem> puts them back together.
For a concrete example, consider the <trigonometric polynomial>
$$
f(t)=\cos(2\pi t)+\frac12\cos(6\pi t)+\frac14\cos(10\pi t).
$$
With $t$ measured in seconds, its three <Fourier harmonics> have <frequencies> $1$, $3$, and $5$ cycles per second and <amplitudes> $1$, $1/2$, and $1/4$. Each <cosine> has zero <wave phase>. Adding them creates the sharper peaks in the figure. The <frequency spectrum> makes the recipe immediately visible; changing an <amplitude> changes the shape without changing the component's <frequency>. <Fourier orthogonality> lets us recover each <Fourier coefficient> by averaging against the corresponding <sine> or <cosine>, because the other <Fourier harmonics> cancel over a full <period of a function>.
\Image[/fourier-frequency-decomposition.png]
{title=Three Fourier harmonics, their sum, and their amplitude spectrum}
The deeper reason these components are so useful is that they simplify operations. By <Euler's formula>, the <exponential function> $e^{i\omega x}$ packages a <cosine> and a <sine> into one <Fourier mode>. A <translation> or a <derivative> changes only its multiplier:
$$
e^{i\omega(x-a)}=e^{-i\omega a}e^{i\omega x},\qquad \frac{d}{dx}e^{i\omega x}=i\omega e^{i\omega x}.
$$
Thus <Fourier modes> are simultaneous <eigenfunctions> of these operations. In <Fourier series>, this gives a literal <diagonalization> in the <Fourier basis>; on the real line, the <Fourier transform> gives the continuous counterpart, with each <frequency> handled separately. The <Translation property of the Fourier transform> and the <Fourier transform of a derivative> turn shifting and differentiating a whole <function> into multiplication of its <Fourier transform>. A <convolution> likewise becomes multiplication by the <convolution theorem>. This is beautiful because one description simplifies several apparently different operations at once.
For example, the <heat equation> $u_t=\kappa u_{xx}$, with <thermal diffusivity> $\kappa>0$, evolves a spatial <Fourier mode> as
$$
u(x,t)=A e^{-\kappa\omega^2t}e^{i\omega x}.
$$
Fine ripples have larger <angular frequencies> and decay faster: the formula explains why heating smooths a jagged temperature profile. For a general initial profile with sufficient regularity, the <Fourier transform> gives $\widehat u(\omega,t)=e^{-\kappa\omega^2t}\widehat u(\omega,0)$, and <Fourier inversion> recovers the temperature. Similarly, multiplying selected parts of a <frequency spectrum> suppresses unwanted components. Claude E. Shannon's \i[Communication in the Presence of Noise] (1949), Section III, describes ideal filtering as a projection that retains some <frequency> coordinates and removes others.https://fab.cba.mit.edu/classes/S62.12/docs/Shannon_noise.pdf[Shannon 1949, Section III, p. 13: filtering in frequency coordinates]{ref}
The physical motivation is already visible in Joseph Fourier's \i[Théorie analytique de la chaleur] (1822). Chapter III constructs heat profiles using sums of <sines> and <cosines>; Chapter IX, articles 355–359, passes from <Fourier series> to <integrals> as the spatial interval becomes unbounded. These heat problems explain the motivation for the discrete and continuous descriptions. Their modern formulations specify which <functions> admit reconstruction and in what sense the expansions converge.https://fr.wikisource.org/wiki/Th%C3%A9orie_analytique_de_la_chaleur/Chapitre_3[Fourier 1822, Chapter III: trigonometric representations of heat profiles]{ref}https://fr.wikisource.org/wiki/Th%C3%A9orie_analytique_de_la_chaleur/Chapitre_9[Fourier 1822, Chapter IX, articles 355–359: from series to integrals]{ref}
Back to article page
