There is a quote fairly widely attributed to Fourier, but I can't substantiate it. That is, I can't verify that he actually said or wrote it (in any language). Can anyone help me out? Here is the quote in English: "An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids". For example, the statement appears in Gao and Yan (2011): It seems it has been attributed both to Fourier's 1807 dissertation and to his 1822 "Analytical Theory of Heat" monograph. The 1822 work has long ago been translated into English by Alexander Freeman and is available on archive.org. However, I can't find any text written by Fourier that is a close match to the claimed quote. Anyone have a solid reference (in any language)?
I have already asked what is basically this same question on math.stackexchange, where it was suggested that I ask it here on hsm.
Of special interest in the claimed quote is the explicit phrase, "continuous or with discontinuities" (even though this can be inferred implicitly from "fonctions etièrement arbitraires"). The explicit reference to both continuous and discontinuous was, historically, a revolutionary claim. Before the Fourier expansions with coefficients calculated using integrals and bases of sinusoids, there was the Taylor expansions with coefficients calculated using derivatives and bases of polynomials $(x-a)^n$. The Taylor expansions were limited to analytic functions...and thus global expansions of discontinuous functions were simply out of the question. Then along came Fourier with a truly revolutionary claim that his expansion worked not only for continuous functions, but discontinuous as well! And to add insult to injury, Fourier wasn't even really a mathematician (but rather an engineer among other things). For more information, see Enders A. Robinson's paper, "A Historical Perspective of Spectrum Estimation"