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A couple days ago I posted this on MSE (here) but in retrospect it might be more appropriate for this site.

This theorem is well-known (maybe it can be called Morera's theorem):

A continuous function satisfying the mean value property on balls is harmonic.

I was recently surprised to hear in a talk that the conclusion still holds if you only check the mean value property on three (I think) radii. Does anyone have a reference or name for this result? I would enjoy seeing the details and a proof.

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  • $\begingroup$ You mean that a function is defined on the whole $\mathbb{R}^n$ and there exist three positive radii such that mean value property holds for all balls of these radii? $\endgroup$ – Fedor Petrov Jul 10 '16 at 18:43
  • $\begingroup$ @FedorPetrov I actually don't know, the speaker was very vague. $\endgroup$ – Funktorality Jul 10 '16 at 19:44
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Yes, in fact you only need two radii. More precisely, a theorem on page 167 of this Monthly paper of Zalcman says any two radii $r_1$ and $r_2$ will work unless the quotient $r_1/r_2$ is a quotient of zeros of a certain explicit function. The author says this result "was discovered by Jean Delsarte as far back as 1957."

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  • $\begingroup$ I haven't read the original sources, but Zalcman's paper says that for $n = 1$ it suffices for $r_1/r_2$ to be irrational. I quote from Zalcman: "there are, for each $n > 1$, at most a finite number of excluded ratios. (The case $n = 1$ is special: (9) reduces to $\cos z - 1$, and the exceptional set consists of the rationals.) When $n = 3$, no ratios need be excluded [9]. Whether the exceptional set is nonempty for any $n > 1$ remains an open question." $\endgroup$ – Dave Witte Morris Jul 10 '16 at 19:28
  • $\begingroup$ That makes sense. The obvious counterexamples (in $d=1$) are something like $r_j=j$. $\endgroup$ – Christian Remling Jul 10 '16 at 19:50
  • $\begingroup$ Thanks for the reference. It appears the speaker misstated the number of radii and the restriction on their ratios! $\endgroup$ – Funktorality Jul 10 '16 at 19:54
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I guess this is the theorem :

Let $f$ be an infinitely differentiable function defined in $\mathbb R^{n}$, and $u(x,r)$ the mean value of $f$ taken over the sphere with center at $x$ and radius $r$, and let $a$ and $b$ denote two fixed positive numbers. If $u(x,a)=u(x,b)=f(x)$ in $\mathbb R^{n}$, then $f$ is harmonic. When $n>3$, exception must be made of a finite number of ratios $a/b$ which are independent of the function $f$.

The theorem is proved in :

J. Delsarte, J.-L. Lions, Moyennes généralisées, Comment. Math. Helv. 33 1959 59–69.

also in the lecture notes (see in particular Chapter III) :

J. Delsarte, Lectures on topics in mean periodic functions and the two-radius theo- rem, Tata Institute, Bombay, 1961. http://www.math.tifr.res.in/~publ/ln/tifr22.pdf

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