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If we look at the 1D heat equation on a conducting rod with non-insulated ends, we get the standard Heat Equation from which the Fourier series formula is derived. I know that the heat equation had standard solutions that were known at the time, which were an exponential multiplied by a sinusoid, and by using separation of variables one can find the exact structure of all of these solutions.

My question is, how did Fourier realize that an infinite number of weighted sines were required to express the general solution? I understand that he knew due to the linearity of the PDE a linear sum of sines would also be a solution, but how did he jump from that to an infinite weighted sum of sinusoids as the representation of the general solution?

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English translation of Fourier's Analytical Theory of Heat is accessible on Internet Archive. His reasoning for a stationary plate problem, which is his first example of applying trigonometric series, is on pp. 133-137. He separated the variables in the Laplace equation and showed that $e^{-mx}cos(my)$ solves it for any $m$. Superposition principle then suggests that one should look for a general solution as a weighted sum of all "elementary" ones. Moreover, in examples, he saw that infinitely many $m$ can be needed to satisfy the boundary conditions:

"In order to consider the problem in its elements, we shall in the first place seek for the simplest functions of $x$ and $y$, which satisfy equation (a); we shall then generalise the value of $v$ in order to satisfy all the stated conditions. By this method the solution will receive all possible extension, and we shall prove that the problem proposed admits of no other solution... A more general value of $v$ is easily formed by adding together several terms similar to the preceding, and we have $$v = ae^{-x}\cos y + be^{-3x}\cos 3y + ce^{-5x}\cos 5y + de^{-7x}\cos 7y + \&c... ...... (b).$$ It is evident that the function $v$ denoted by $\phi(x, y)$ satisfies the equation $\frac{d^2v}{dx^2}+\frac{d^2v}{dy^2} = 0$, and the condition $\phi(x, \pm\frac12\pi) = 0$. A third condition remains to be fulfilled, which is expressed thus, $\phi(0,y)=1$, and it is essential to remark that this result must exist when we give to y any value whatever included between $-\frac12\pi$ and $+\frac12\pi$... Equation (b) must therefore be subject to the following condition: $$1 = a\cos y + b\cos 3y + c\cos 5y + d\cos 7y + \&c.$$ The coefficients, $a, b, c, d, \&c.$, whose number is infinite, are determined by means of this equation... In this manner an exact idea might be formed of the movement of heat in the most general case; for it will be seen by the sequel that the movement is always compounded of a multitude of elementary movements, each of which is accomplished as if it alone existed."

He proceeds to determine the (infinitely many) coefficients recursively.

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What we assume to be "the general solution" has evolved and refined over time. There is an argument to be made that Fourier did not strictly know that this would give the general solution, as the correct function space to define the space of solutions was not known to him. This would be resolved by Laurent Schwartz in the 1940s. Today we call the "fundamental solution" of a linear PDE the distributional impulse (Dirac delta) solution of the PDE, and general (now well-defined) solutions can be recovered by convolution in the function space of tempered distributions (see for example M. Shubin, "Invitation to partial differential equations" AMS (2022).) The Fourier Transforms of the fundamental solution will indeed require frequencies approaching infinity.

Incidentally, the proposal that an infinite sum of sinusoids might be the general solution of a partial differential equation goes back to D. Bernoulli in the context of discussion of the wave equation (Bernoulli, D. (1753). Reflexions et Eclaircissemens Sur les Nouvelles Vibrations des Cordes. Hist. de l’Acad. Roy. de Berlin, 9, pp. 147-195. Reproduction of the original with Portugese translation)

Fourier mentions the problems of determining Bernoulli's coefficients (Analytic Theory of Heat, paragraph 230) and proposes that his method allows for their general computation. In this context he notes that continuity is indeed not a requirement for functions to be amenable to computations of Fourier coefficients. Dealing with discontinuities was not fully resolved until Schwartz.

In this sense one could argue that Schwartz proved Bernoulli, Fourier, Heaviside and others right, by giving the precise setting in which their work is correct. And since then we strictly know, at least with respect to rigorous mathematical formalism.

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