Heat Equation Existence of Fourier Series

Question

I'm currently doing a bit of digging with the Heat Equation and the Fourier Series.

It seems that the boundary condition $u(x,0)=f(x)$ can be arbitrary. At some point, we get something like (in a greatly simplified manner) $f(x)=\sum_{n\geq0}B_n \sin \left ( knx \right )$ for some constant $k$.

Since $f(x)$ is any arbitrary function, does this mean that we've essentially 'proven' that any (non-pathological I guess) function $f(x)$ can be expressed in the form $f(x)=\sum_{n\geq0}B_n \sin \left ( knx \right )$?

Or is this moreso a 'demonstration' or 'motivation' to believe that it is possible that any function can be expressed as the infinite sum of sines and cosines?

Answer 1

Joseph Fourier claimed this in 1822 here:

• Théorie analytique de la chaleur (The analytic theory of heat)

Il résulte de tout ce qui a été démontré dans cette section, concernant le développement des fonctions en séries trigonométriques, que si l'on propose une fonction $f x$, dont la valeur est répresenté dans un intervalle déterminé, depuis $x=0$ jusqu'à $x=X$, par l'ordonné d'une ligne courbe tracée arbitrairement on pourra toujours développer cette fonction en un série qui ne contiendra que les sinus, ou les cosinus, ou les sinus et cosinus des arcs multiples, ou les seuls cosinus des multiples impairs. On ne peut résoudre entièrement les questions fondamentales de la théorie de la chaleur, sans réduire à cette forme les fonctions qui représentent l'état initial des températures.

(It follows from what has been shown in this section, concerning the development of functions in trigonometric series, if one offers a function $f x$, whose value is represented in a given interval from $x = 0$ to $x = X$, ordered by a curved line drawn arbitrarily we can always expand this function in a series that will contain only sine or cosine or sine and cosine multiple arcs or cosine only odd multiples . One can not fully solve the fundamental problems of the theory of heat, without reducing to this form the functions that represent the initial state of temperature.)

(part 235., page 258)

Answer 2

It's implicit in the proof that the series converges in the appropriate sense to $f$; here we require, by the way, that $f$ be defined on some appropriate finite interval $[a, b]$ (we might also allow cosine terms depending on the formulation). Here, the appropriate notion of nonpathological at least requires that we can integrate $f$ against the functions $\sin(knx)$; more precisely, we demand that $f \in L^2([a, b])$; this is also the appropriate space for convergence to a function of its Fourier series. It's not surprising that $L^2$ is a good setting for this problem: The coefficients of the series are given by integrals, which themselves don't see the difference between two functions which differ on a set of measure zero.