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In his answer to a previous question Alexandre Eremenko pointed out that Joseph Fourier in his book Analytic Theory of Heat gave all kinds of arguments in favor of the following mathematical discoveries,

  1. Every periodic function can be expanded into a Fourier series.
  2. Every reasonable function on the real line can be represented by a Fourier integral.

including experimental evidence with heated metal rings.
Can someone provide more details on these heated metal rings and their link with the statements above, as well as a precise reference in Fourier's book?

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    $\begingroup$ Fourier did use experiments with metallic rings (and bars, spheres and cubes) to confirm physical assumptions and illustrate computational effectiveness of his theory, see sections 64 and 109 of his Analytic Theory of Heat. But, as you can see, it was not to argue the more abstract points such as 1 and 2, so no. $\endgroup$
    – Conifold
    Nov 19, 2020 at 10:29
  • $\begingroup$ @AlexandreEremenko Mathematical abstractions are not among the assumptions that are either confirmed or disconfirmed by experiments. Outcomes of experiments testing a theory that uses arithmetic have no bearing on arithmetic itself, 1+1=2 even though water drops can merge. Abstract mathematical theorems, like representation of functions by trigonometric series, are not analogous to physical laws, like the inverse square law of gravity. And Fourier is very clear that it is the latter that his experiments are meant to confirm. $\endgroup$
    – Conifold
    Nov 19, 2020 at 20:42

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The essential part of the answer (page references) is contained in the comment of @Conifold. However his general conclusion is plain wrong, and I would like to put the things straight.

A scientist makes ASSUMPTIONS. Then develops a theory. And then compares with observations/experiments. If this comparison works, this CONFIRMS his assumptions.

For example, Newton (and others) assume the inverse square law. Then Newton (and others) derived a lot of consequences from it, which can be tested by observations. And the agreement with observations proves the initial conjecture. This is how science works.

Returning to Fourier. Of course, he could not prove mathematically the statement that "arbitrary periodic function has a Fourier expansion", for the simple reason that the modern notion of "arbitrary function" did not exist at that time. It was first stated by Dirichlet, whose purpose was to give a mathematical justification of Fourier discoveries.

(Further attempts in this direction led to further evolution of the notion of function: "generalized functions" or "distributions" were also introduced ith the purpose to justify Fourier analysis.

Fourier himself was a scientist, first of all. And he lived at the time when science was not separated from mathematics (it is still not separated completely). So his approach is that of a scientist: he makes assumptions, develops a theory and then tries to test it.

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