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In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.

I want to ask if there is a modern version of this proof exist in literature?

If someone can provide me with an elegant compact proof or reference for it , it would be helpful.

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    $\begingroup$ The proof I know uses the integral between 0 and 1 of $tJ_\nu (zt) J_\nu (\bar z t)dt$ and works for $\nu \geq -1$. I don't know if this is the original one by Fourier. $\endgroup$
    – Giorgio Metafune
    Commented Oct 1, 2022 at 12:57
  • $\begingroup$ This page has the simple proof referred to by @GiorgioMetafune. A more conceptual argument comes from observing that $J_0(x)$ can be obtained as a limit of rescaled Laguerre polynomials (as discussed, e.g., here). The Laguerre polynomials are a family of orthogonal polynomials and therefore have only real zeros. $\endgroup$
    – Dan Romik
    Commented Oct 2, 2022 at 4:19
  • $\begingroup$ (See also this answer to another question, which elaborates on the analogy between Bessel functions and orthogonal polynomials.) $\endgroup$
    – Dan Romik
    Commented Oct 2, 2022 at 4:29
  • $\begingroup$ @DanRomik thank you for the references! $\endgroup$
    – TPC
    Commented Oct 2, 2022 at 9:08
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    $\begingroup$ It is exactly the proof contained in the link provided by @DanRomik $\endgroup$
    – Giorgio Metafune
    Commented Oct 2, 2022 at 10:16

1 Answer 1

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Fourier proof was incomplete. Fourier used the following

Statement. A real entire function has only real zeros if its derivatives have the following property: If $x$ is a real root of $f^{(n)}$ then $f^{(n-1)}(x)f^{(n+1)}(x)<0.$

But he verified this only for polynomials. Fourier's proof was criticized by Cauchy, and Fourier defended his arguments, but the statement above is not correct for unrestricted entire functions. The full story is described in the paper of Pólya, Some problems connected with Fourier's work on transcendental functions, Quart. J. Math. 1 (1930) 21-34.

In this paper he stated what is called since then the “Fourier–Pólya Conjecture”: If $f$ is a real entire function of genus $0$, then the number of points where $f^{(n)}(x)=0$ but $f^{(n-1)}(x)f^{(n+1)}(x)>0$ is equal to the number of pairs of non-real conjugate zeros.

This conjecture was proved only in 2000:

H. Ki and Y. Kim, On the number of non-real zeros of real entire functions and the Fourier–Pólya conjecture, Duke Math. J. 104 (2000) 45–73.

(The earlier paper of Kim

Kim, Young-One, Critical points of real entire functions whose zeros are distributed in an infinite strip, J. Math. Anal. Appl. 204 (1996), no. 2, 472–481.

already contained what is needed to justify Fourier's argument).

Thus Fourier's method has been justified, after 170+ years of research, but it is incomparably more complicated than the "modern" proof based on the observation that zeros of Bessel's function are eigenvalues of a self-adjoint operator, and these eigenvalues are real by a two-line linear algebra argument. If I remember correctly, this observation is due to Poisson.

See also my lectures with an exposition of other related conjectures of Pólya: https://www.math.purdue.edu/~eremenko/dvi/kent.pdf

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    $\begingroup$ Ah very nice, I did not know at all. This means that the proof I mentioned in the comment above is due to Poisson? $\endgroup$
    – Giorgio Metafune
    Commented Oct 1, 2022 at 13:11
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    $\begingroup$ @Giorgio Metafune: Yes. This "modern elegant proof" of the fact that eigenvalues of an Hermitian operator are real is due to Poisson, though I do not remember the exact reference. $\endgroup$
    – Alexandre Eremenko
    Commented Oct 1, 2022 at 13:46
  • $\begingroup$ @AlexandreEremenko Thank you very much for such a detailed answer with historical context. Helped me clarify my confusion about the subject.( As everyone credited Fourier for the theorem but couldn't find a proof by him) $\endgroup$
    – TPC
    Commented Oct 2, 2022 at 8:58
  • $\begingroup$ Fascinating story. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Oct 2, 2022 at 17:38

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