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Fourier stated that every function can be decomposed into sine and cosine functions. Was he referring to periodic functions only? To a certain class only? I ask, because it seems clear (at least to me) that most functions cannot be so decomposed. One simple example is the n-th degree polynomial. A small portion can be approximated, but the whole function finds no place in the function space spanned by the sines and cosines.

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    $\begingroup$ Fourier was allowing for infinite sums of sines and cosines. $\endgroup$ Commented Jul 18, 2021 at 20:19
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    $\begingroup$ Fourier's work was before the realization/acceptance of our present conception of an arbitrary function, and thus general statements such as "all functions" have to be read in historical context. Rather than extract isolated statements by Fourier (possibly not even exactly quoted) and try and interpret them from our present thinking, perhaps look at what he actually wrote (at least a translated version): The Analytical Theory of Heat, translated with notes by Alexander Freeman, Cambridge University Press, 1878, xxiii + 466 pages. $\endgroup$ Commented Jul 18, 2021 at 21:55
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    $\begingroup$ I think at this point you need to give an exact and specific reference to the statement of Fourier's that concerns you, which should be possible because virtually everything published in Latin or a Romance language during the 1800s is freely available on the internet. $\endgroup$ Commented Jul 19, 2021 at 8:22
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    $\begingroup$ Regarding the notion of "whole function", keep in mind that the precise notion we have today differs from Fourier's time in that now we insist that a specific domain and codomain be part of what a function is. Besides, if you want to analyze the heat in an object, behavior light years away is not particularly relevant. And if it were, then you'd break the analysis into separate regions for analysis, even when studying the structure of the universe as a whole (e.g. manifolds). See also this answer. $\endgroup$ Commented Jul 19, 2021 at 10:13

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First of all, one has to notice that the work of Fourier predates all modern notions of function. The notion of function was first clarified by Dirichlet, in his attempt to justify Fourier's arguments. What Fourier probably meant was that

a) any reasonable periodic function, say with period $2\pi$ can be expanded into a series of the form $\sum a_k\cos kx+b_k\sin kx$. And

b) that many non-periodic functions on the real line can be represented by Fourier integrals $$\int \cos(sx)\phi(s)ds,\quad\int\sin(sx)\psi(s)ds.$$

Fourier was solving specific problems of mathematical physics. Even his contempories understood that many of his arguments are not mathematically rigorous (and he had difficulty with publication of his book for this reason; it was very much criticized).

He did not specify the exact class of functions for which these statements are true, and it took mathematicians many decades to arrive at exact statements. The very notion of function evolved in this process, beginning with Dirichlet's definition, and proceeding to "$L^2$-functions" (which are not exactly functions in the Dirichlet sense), and then to distributions and hyperfunctions. All these notions were invented with the purpose of making precise sense of Fourier ideas.

To take your specific example of polynomials, only in the 1950s it became clear how to represent polynomials as Fourier integrals; this requires the theory of "tempered distributions".

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  • $\begingroup$ "Tempered distributions" Almost poetry. :) $\endgroup$ Commented Jul 20, 2021 at 8:02

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