Fourier series

Recall that a function f : $ℝ$ → $ℂ$ is said to be T – periodic if,

for all real x ;

We can always go back to T = 2 π, even if it means considering g(x) = $f\left(\frac{\mathit{Tx}}{2\mathrm{\pi }}\right)$ .

One of the essential properties of the number π is that the functions cos nx, sin nx, e ^inx (where n is an integer), known as "elementary signals", are 2 π - periodic, as are their linear combinations. A natural question then arises: do we obtain all 2 π functions – periodic in this way? We'll see that the answer is essentially "yes" if we allow infinite linear combinations (series), but tricky problems of regularity of the function f and convergence of the series arise.

Fourier series theory, initiated by Fourier in his Théorie analytique de la chaleur, originally had a similar aim: to show that all solutions of a certain partial differential equation, known as the heat equation (we'll study this in the applications), can be obtained as a superposition of elementary solutions; today, the aim of this theory is to specify how a more or less arbitrary periodic function f 2 π – can be obtained from elementary signals, and conversely to see the functions f that can be obtained by taking more or less arbitrary infinite linear combinations of elementary signals, say ${\displaystyle \sum c_n{\mathrm{e}}^{\mathrm{i}\mathit{nx}}}$ :

• the first operation is called the analysis of f ;

• the second the synthesis of c _n e ^inx signals.

These two operations are inverses of each other, like derivation and integration.

Consider the example of a Poisson core P _...