# substantiating Fourier quote about “an arbitrarily capricious graph”

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): http://books.google.com/books?vid=ISBN1441915451&pg=PA19 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 at http://www.archive.org/details/analyticaltheory00fourrich However, I can't find any occurrence of "capricious" or anything else closely similar to the quote in A. Freeman's translation. Anyone have a solid reference (in any language)?

Let me quote Fourier's "Théorie de la chaleur", p. 556

La suite trigonométrique égale au second membre est convergente; le sens de cette dernière proposition est que, si l'on donne à la variable $$x$$ une valeur quelconque, la somme des termes de la suite s'approche de plus en plus, et infiniment près, d'une limite déterminée. C'est cette limite qui est $$0$$, si l'on a mis pour $$x$$ une quantité comprise entre $$0$$ et $$X$$, mais non comprise entre $$a$$ et $$b$$; et si cette quantité mise pour $$x$$ est comprise entre $$a$$ et $$b$$, la limite de la série a la même valeur que $$fx$$. Cette dernière fonction n'est assujettie à aucune condition, et la ligne dont elle représente l'ordonnée peut avoir une forme quelconque; par exemple, celle d'un contour formé d'une suite de lignes droites et de lignes courbes.

According to DeepL, the last sentence translates as

The latter function is not subject to any conditions, and the line for which it represents the $$y$$-axis may have any shape; for example, that of a contour formed by a series of straight lines and curved lines.

Thus it looks like an arbitrarily capricious graph is a free translation of a fonction assujettie à aucune condition (a function not subject to any conditions). However, a curved line probably meant a continuous curved line for Fourier.

Maybe relevant, Théorie (1822), §220 :

even entirely arbitrary functions [fonctions etièrement arbitraires] may be developed in series of sines of multiple arcs.

Thus, we have "arbitrarily" but not "capricious", but IMO the key word here is arbitrary.

The issue is that the general concept of function evolved slowly and for many Early Modern authors a function was implicitly continuous (given by a law).

The modern notion of "functional dependence" is credited with Dirichelet (1829).

But we can find it already into Fourier's Théorie (1822); see §417 :

In general the function $$f(x)$$ represents a succession of values or ordinates each of which is arbitrary. [...] We do not suppose these ordinates to be subject to a common law.

It may follow from the very nature of the problem [...] that the passage from one ordinate to the following is effected in a continuous manner. But [...] the general equation [representing the function by an integral] is independent of these conditions. It is rigorously applicable to discontinuous functions [emphasis added].

Thank you everyone for your helpful responses---including the suggestion to ask on hsm, which I will probably do. It seems the claimed quote captures the essence of Fourier 1822, but yet does not match closely enough to warrant the quote marks. One key component that is missing is the explicit phrase, "continuous or with discontinuities" (again, 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 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" at http://www.archive.ece.cmu.edu/~ece792/handouts/Robinson82.pdf

• I have asked what is basically this same question on <a href="hsm.stackexchange.com/questions/11172/…>, as suggested by @kimchi-lover. – Daniel J. Greenhoe Nov 20 '19 at 11:10
• Good point ! The issue is that the "general" concept of function evolved slowly and for many ancient authors a function was implicitly continuous (given by a law). The modern notion of "functional dependence" is credited with Dirichelet (1829). – Mauro ALLEGRANZA Nov 20 '19 at 15:11