The following quote (in English) is fairly widely attributed to Fourier, but I can't substantiate it.

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.

That is, I can't verify that he actually said or wrote it (in any language). Can anyone help me out?

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 SE, where it was suggested that I ask it here on HSM SE.

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".

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    $\begingroup$ Are we generally confident that Fourier used the term "discontinuous" for the same type of functions we modern folk call "discontinuous"? Not denying; just that I don't know. $\endgroup$ – Carl Witthoft Nov 20 '19 at 13:30
  • $\begingroup$ I think that Carl Witthoft pinpointed the real question: do Fourier proved that a discontinouos function (in the modern sense) can always be expressed as a sum of sinusoids? In his dissertation Fourier don not provide a definition of discontinuous function (and of course of continuous function) in a point or in an interval, so we can only infer such a definition from the context. In article 14 we read: "fonctions qui ne sont point assujéties à une loi constante, et qui représentent les ordonnées des lignes irrégulières ou discontinues", so ... $\endgroup$ – user6530 Nov 21 '19 at 14:46
  • $\begingroup$ we can deduce that a function like f(x)=x^2*sin(1/x) for x!=0 and f(0)=0 is not continuous (nor differentiable) in the Fourier's sense. $\endgroup$ – user6530 Nov 21 '19 at 14:49
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    $\begingroup$ About continuity and discontinuity you can read here jstor.org/stable/pdf/2325087.pdf some interesting observations (see page 8(433)) and if you can read french this arxiv.org/ftp/arxiv/papers/1211/1211.0794.pdf especially the quot of Libri "... fonction discontinue quelconque, dont les diverses parties, comprises entre des limites données de la variable, suivent une marche dissemblable, et sont représentées par des expressions différentes" where it is clear that actually Fourier considered defined piecewise continuous functions with a finite number of discontinuities $\endgroup$ – user6530 Nov 25 '19 at 11:51
  • $\begingroup$ The jstor one is also available here: math.ucdavis.edu/~saito/courses/121/gonzalez.pdf $\endgroup$ – Daniel J. Greenhoe Nov 27 '19 at 10:53

You can find the quote (or something similar) in multiple places in the Théorie analytique de la chaleur, for example in

Chapter I,

paragraph (article) 14:

"L'examen de cette condition fait connaître que l'on peut développer en séries convergentes, ou exprimer par des intégrales définies, les fonctions qui ne sont point assujéties à une loi constante, et qui représentent les ordonnées des lignes irrégulières ou discontinues. Cette propriété jette un nouveau jour sur la Théorie des équations aux différences partielles, et étend l'usage des fonctions arbitraires en les soumettant aux procédés ordinaires de l'analyse."

Chapter IV, Section VI,

paragraph 219:

"Nous avons supposé jusqu'ici que la fonction dont on demande le développement en séries de sinus d'arcs multiples, peut être développée en une série ordonnée, suivant les puissances de la variable x, et qu'il n'entre dans cette dernière série que des puissances impaires. On peut étendre les mêmes conséquences à des fonctions quelconques, même à celles qui seraient discontinues et entièrement arbitraires."

paragraph 220:

"Cette remarque est importante, en ce qu'elle fait connaître comment les fonctions entièrement arbitraires peuvent aussi être développées en séries de sinus d'arcs multiples."

paragraph 229:

"Les suites formées de sinus ou de cosinus d'arcs multiples sont donc propres à représenter entre des limites déterminées, toutes les fonctions possibles, et les ordonnées des lignes ou des surfaces dont la loi est discontinue."

paragraph 278:

"[...] Ainsi ce théorême qui donne, entre des limites assignées, le développement d'une fonction arbitraire en séries de sinus ou de cosinus d'arcs multiples se déduit des règles élémentaires du calcul. [...] Dans la question suivante, on réduit encore la fonction arbitraire en une série de sinus"


After some search, I think I've finally found the source of the quote (which is not Fourier).

The phrase appear almost verbatim at the very beginning of Cornelius Lanczos's Discourse on Fourier Series (1966):

In a memorable session of the French Academy on the 21st of December 1807, the mathematician and engineer Joseph Fourier announced a thesis which inaugurated a new chapter in the history of mathematics. [...] Fourier claimed that an arbitrary function, defined in a finite interval by an arbitrary capricious graph, can always be resolved into a sum of pure sine and cosine functions.

Now the questions are: is this reconstruction historically correct? Is Fourier's claim reported accurately?

First of all, it's really hard to say that the audience considered that session to be truly memorable. The procès-verbal of the seat is exactly the following:

M. Fourier lit un Mémoire sur la Propagation de la chaleur sur le solides. MM: Lagrange, Laplace, Monge et Lacroix.


Fourier reads a Memoir on Heat Propagation on Solids. Lagrange, Laplace, Monge et Lacroix [were present].

Subsequently, a rather cold account, written by Poisson and simply signed by "P.", appears in March 1808 in the Bulletin des Sciences, where he mentions the heat equation but not the "Fourier" series.

Anyway, now we know, following Lanczos, that the claim was made by Fourier either orally during the seat of the French Academy or written in his original 1807 thesis. The former hypothesis should be discarded, as there are no detailed records of the session, and the latter must be rejected because in the Théorie de la propagation de la chaleur dans les solides, published only in 1822 under the name of Théorie analytique de la chaleur, there's no occurrence of terms like "capricieux" or "graph(e)".

So, considering also the anachronism and the inaccuracy of a sentence such as "[a] function [...] defined [...] by an arbitrarily capricious graph", we must conclude that @Daniel J. Greenhoe's answer is correct:

Possibly what happened was that someone wrote a commentary/summary regarding Fourier's work, and then later someone else came along thinking it was an actual quote [...]

Moreover, when Lanczos want to quote exact words, also in translation, he uses quotation marks, as in the following (p. 8):

The gradual clarification of our ideas concerning the concept of a function has much to do with the discovery of Fourier in 1807. Fourier was well aware of the fact that a function need not be everywhere differentiable in order to be representable by the sum of an infinite sequence of sine and cosine functions. He talk of "arbitrary function" $y=f(x)$ which could represented by his series. The lack of precision in his language was heavily criticised by his contemporaries, although in fact Fourier came nearer to the truth than his critics, who eanted to restrict the validity of the Fourier series to the class of infinitely differentiable functions.

Here quotation marks around arbitrary function are correct, as Fourier actually uses the expression fonctions entièrement arbitraires.

We must not blame Lanczos for this, and himself was well aware that his book was not was not a faithful account of the history of Fourier series, as reported in the foreword of the 2016 edition:

His book is not dry; a person inhabits it. "And why is it not allowed," he says in the video, "to put in a certain emotional emphasis? If you're enthused about something, why not say so?"

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  • $\begingroup$ None of your quotes says "capricious", but merely "completely arbitrary", right? $\endgroup$ – Gerald Edgar Nov 23 '19 at 14:10
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    $\begingroup$ @GeraldElgar No, the term "capricious" never appear in the Théorie analytique de la chaleur nor in the second part of the Théorie de la propagation de la chaleur dans les solides (from article 80 on, the first part in another volume that I cannot consult in this moment), so I totally agree with Daniel J. Greenhoe. Anyway, I think that there are some evidences that the quote is at least altered, as I'm sure that Fourier never said something like "function defined in a finite interval" but used that phrase "développement entre des limites déterminées", i.e. "between some defined boundaries" ... $\endgroup$ – user6530 Nov 25 '19 at 11:42

I think the conclusion here is that there is no proof that the claimed Fourier quote is an actual Fourier quote (in any language). However,
(1) Absence of proof is not proof of absence.
(2) Although Fourier may have not communicated the quote in substance, he did arguably communicate the quote in essence.

Possibly what happened was that someone wrote a commentary/summary regarding Fourier's work, and then later someone else came along thinking it was an actual quote, published the commentary with quotes around it... and thus a mathematical urban legend was born(?).

Any author who is looking for a substantiated Fourier quote to replace the claimed quote could consider extracting one from Fourier 1822 Section 219, as referenced by @user6530:
"Up to this point we have supposed that the function whose development is required in a series of sines of multiple arcs can be developed in a series arranged according to powers of the variable x,... We can extend the same results to any functions, even to those which are discontinuous and entirely arbitrary."
(Fourier (1822), §219, page 184, translation by Alexander Freeman).

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