3

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.

Rodrigo de Azevedo
  • 1
  • 9
  • 15
Deschele Schilder
  • 1
  • 8
  • 14
  • 2
    Fourier was allowing for infinite sums of sines and cosines. – Dave L Renfro Jul 18 '21 at 20:19
  • @DaveLRenfro But you cant expand f=x in an infinite amount of them. – Deschele Schilder Jul 18 '21 at 20:22
  • 2
    See Fourier Series of $f(x) = x$ AND Fourier: $f(x) = x$ with $0 \leq x \leq 2\pi$ (Exercise almost solved) AND Compute the Fourier series for $f(x)=x$ over the interval $-\pi\leq x \leq\pi$ AND Fourier series expansion for $f(x)=x$ over the interval $-\pi\leq x\leq \pi$ AND Fourier series of $f(x)=1$ and $f(x)=x$. – Dave L Renfro Jul 18 '21 at 20:50
  • @DaveLRenfro But thats not the whole function. Only a small part. – Deschele Schilder Jul 18 '21 at 21:04
  • The expansions are for an interval given in advance. In fact, the Fourier coefficients are in terms of definite integrals whose integration limits are the endpoints of the interval. Also, many functions can't be expanded (exactly, at each point) throughout a given interval for various reasons, one of which is that any such expansion is the limit of continuous functions, and as such is continuous at lots of points (see this answer and this exposition paper). (continued) – Dave L Renfro Jul 18 '21 at 21:45
  • 2
    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. – Dave L Renfro Jul 18 '21 at 21:55
  • @DaveLRenfro Didnt he know the function f(x)=x? How is that function different in a historical context? – Deschele Schilder Jul 18 '21 at 22:05
  • 1
    Maybe look at what is involved in a Fourier expansion. The basic computations and ideas require nothing more than U.S. 2nd semester calculus. For instance, the functions one deals with are defined on a specific interval, often taken to be $[-\pi,\pi],$ and the resulting Fourier series is essentially--there are some technicalities involved--the periodic extension of this function to the real line. I need to leave now. Maybe look at EXACTLY what Fourier said/wrote, rather than rely on second-hand "quotes". – Dave L Renfro Jul 18 '21 at 22:54
  • @DaveLRenfro You are making the common mistake that you can add all of the interval defined functions. – Deschele Schilder Jul 18 '21 at 23:00
  • The Fourier heat function is defined over the whole space. Not on an interval. So Foutier meant by all functions all that are defined on an interval? In that case he is always right obviously. – Deschele Schilder Jul 19 '21 at 01:52
  • 2
    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. – Dave L Renfro Jul 19 '21 at 08:22
  • @DaveLRenfro Well, I have no official source. I saw it mentioned in this question: https://hsm.stackexchange.com/questions/13396/what-does-the-fourier-transform-have-to-do-with-heat – Deschele Schilder Jul 19 '21 at 09:24
  • @DaveLRenfro Likewise, every function can be approximated by powers. But only in the neighborhood of a point you choose. So not the whole function at once. You can say of course f(x)=c+ax+bx^2+dx^3...etc. but the RHS can only be evaluated wrt f. – Deschele Schilder Jul 19 '21 at 10:00
  • 3
    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. – Dave L Renfro Jul 19 '21 at 10:13
  • @DaveLRenfro On page 187 of your reference it is mentioned that the expansion in sines and cosines is valid for only a certain limited interval. So not the whole function. Of course it can be done for every interval. So the whole function can indeed be approached by an infinite sum at every x. But the function cannot be replaced as a whole by sines and cosines. Rather, for each x an approximation can be made by using sines and cosines whose coefficients are based on an integration of the whole function and the cosines and sines. That by itself is no new function unless you limit the x's. – Deschele Schilder Jul 19 '21 at 11:10
  • @DaveLRenfro So the whole function cannot be replaced by a function in which the the function itself is not contained. Only in small intervals this can be done. Obviously it can be done for periodic functions. – Deschele Schilder Jul 19 '21 at 11:13

1 Answers1

7

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".

Alexandre Eremenko
  • 48,930
  • 3
  • 80
  • 177