Determining a function is harmonic from mean value property for just three(?) radii 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.
 A: 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."
A: 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
