TITLE: Did Joseph Fourier ever make a pure mathematical mistake?
QUESTION [44 upvotes]: Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001:

It is remarkable that the nineteenth century was a time of error for
  mathematics: not trivial oversights or amateur conclusions, but
  fundamental mistakes in the understanding of mathematical concepts and
  the formulation of mathematical proofs. These mistakes were not
  restricted to unknown mathematicians but occurred in the works of
  great mathematicians such as Joseph Fourier, Augustin Cauchy, and
  Denis Poisson.

I'm highly interested what are the mathematical mistakes that Fourier did, especially if there are any at the fields that relate to series and transform named after him?

REPLY [5 votes]: In addition to other good answers, it might be worth noting that if we do not insist that functions be "pointwise", and do not insist that convergence of partial sums of Fourier series be pointwise, then there's no difficulty in making a very broad interpretation (not Fourier's original) that "everything is represented by its [Fourier] series" completely legitimate. A starting point is the (originally disturbing-to-me) fact that while Fourier series of continuous functions do converge in $L^2$, there are problems with pointwise convergence. But no problems with $L^2$ convergence even for $L^2$ functions. The simplicity of the Hilbert-space aspects is exploited in ($L^2$-) Sobolev spaces (initiated c. 1906-7 by Beppo Levi and G. Frobenius in the guise of "energy norms", and then systematically by Sobolev in the 1930s)... which leads to the assertion that every distribution on the circle has a Fourier series converging to it in a suitable Sobolev space. Termwise differentiation is always justified (if interpreted distributionally). And so on. 
E.g., the objection that $\sum_{n\in\mathbb Z} 1\cdot e^{2\pi inz}$ does not converge pointwise is essentially irrelevant to the provable fact that it does converge to the periodic Dirac $\delta$ in the Sobolev space $H^{-1/2-\epsilon}$ for every $\epsilon>0$, where $H^s$ is the completion of smooth functions on the circle with respect to the $H^s$ norm defined at first on test functions by
$|f|^2_s=\sum_{n\in\mathbb Z} |\widehat{f}(n)|^2\cdot (1+n^2)^s$.
My interpretation of such possibilities is that, in many applications (both to physical sciences and to more esoteric mathematical situations) the reason Fourier series work so well is that, despite our inherited penchant for worrying about pointwise behavior, it's not pointwise behavior that matters very much, but, rather, various averaged versions.