The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu + 1}\|\mathbf a\|^{-\nu}J_\nu(\|\mathbf a\|), \qquad\nu=\frac n2 -1, \tag1 $$ found e.g. in [1, p. 198] or [2, p. 154].

Does anyone here know earlier references, and perhaps who first published this formula?

According to Watson [3, p. 9] the case n=2, $$ \frac1{2\pi}\int_{S^1}e^{ia\cos\theta}d\theta=J_0(a) \tag2 $$ goes back to Parseval [4], but I am mainly curious about the case n=3, $$ \frac1{4\pi}\int_{S^2}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u)=\frac{\sin\|\mathbf a\|}{\|\mathbf a\|}. \tag3 $$

  1. I. M. Gel'fand & G. E. Shilov, Generalized functions, vol. 1, Academic Press (1964).

  2. E. M. Stein & G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton UP (1971).

  3. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge UP (1922)

  4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

  • $\begingroup$ On my opinion, this is an ill-defined question: who evaluated an elementary integral for the first time. Euler, who introduced $\exp$ could do this, without giving it the name "Fourier transform". Parseval [4] did a more complicated integral... $\endgroup$ – Alexandre Eremenko Nov 23 '13 at 14:05
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    $\begingroup$ @AlexandreEremenko That's exactly what I'd like to know: Do we have evidence that Euler could do (3)? (It's not that elementary...) Then OK, I replaced "first derived" by "first published". Note that Watson or Encykl. Math. Wiss. attribute scores of formulas -- but not this one, as far as I could find. $\endgroup$ – Francois Ziegler Nov 23 '13 at 15:32
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    $\begingroup$ Actually the $n=3$ case is entirely elementary, using a theorem of Archimedes: the projection $(r \cos \theta, r\sin \theta, z) \mapsto (\cos \theta, \sin \theta, z)$ from $S^2$ to the cylinder $\{x^2+y^2=1, \; |z|\leq 1\}$ preserves areas. Thus the integral is just $\frac12 \int_{-1}^1 e^{i\|a\|z} dz$, which Euler certainly knew was ${\mathop {\rm sinc}}(\|a\|)$. $\endgroup$ – Noam D. Elkies Nov 30 '13 at 17:09
  • $\begingroup$ @NoamD.Elkies I agree that the formula is elementary modulo Archimedes, I'd just be surprised to see Poisson state it (below, item 2.) without at least as much of an argument as you have given -- if this was the first time it appeared. Instead, he justifies it by calling it a "known formula". To me this suggests there may be an earlier occurrence, and this "first" is what I'm looking for. But maybe it doesn't exist. $\endgroup$ – Francois Ziegler Dec 1 '13 at 2:11
  • $\begingroup$ $S^n$ is coadjoint orbit, so you can see in general case Kirillov's character formula en.wikipedia.org/wiki/Kirillov_character_formula $\endgroup$ – user21574 May 5 '14 at 7:02

At the risk of answering my own question, here is what I have since found:

  1. For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, Summation of multiple Fourier series by spherical means, Trans. AMS 40 (1936) 175-207. Bochner exposes it again on pp. 73-74 of Fourier Transforms (Princeton UP 1949).

  2. For $n=3$, Burkhardt (Trigonometrische Reihen und Integrale bis etwa 1850, Encykl. Math. Wiss. II A 12 (1916) 819-1354, page 1258) claims to find formula (3) in Poisson's Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques, Mém. Acad. Roy. Sci. Inst. France 3 (1820) 121-176, page 134, in the form $$ \mathfrak{Sin}\,pt= \frac{pt}{2\pi}\int_0^{2\pi}\int_0^\pi\exp\{t(g\cos u+h\sin u\sin v+k\sin u\cos v)\}\sin u\,du\,dv $$ where $p=\sqrt{\smash[b]{g^2+h^2+k^2}}$, $\mathfrak{Sin}$ is a hyperbolic function, and Burkhardt is missing a factor of 2. However... I'm not able to find it on that page of Poisson. On the other hand Poisson states it as "known" in a later memoir (1831, page 558). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?

Edit: Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited Mémoire, but on page 174 instead of 134, in the form $$ \int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv = 2\pi\frac{e^{atp}-e^{-atp}}{atp}. $$

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  • $\begingroup$ I think this question is related to the question of a Fourier invariant (up to scaling) representation of spherical harmonics on the surface of the unit sphere. Something like the Fourier invariant (up to scaling) Hermite polynomials for the surface of the unit sphere. I am not as mathematically sophisticated, but I think the answer is Laguerre polynomials expanded in the usual legendre polynomials? $\endgroup$ – v217 Oct 14 '16 at 9:11

The answer is:

Weyl, H. 1919, Annalen der Physik, 365, 481

doi: 10.1002/andp.19193652104

Though it's difficult to find in there if you don't understand German

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  • $\begingroup$ Thank you. But, unless I missed it, Weyl’s paper only considers the cases n = 2, 3 done before by Parseval and Poisson, no? $\endgroup$ – Francois Ziegler Apr 3 at 21:46

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