A couple days ago I posted this on MSE ([here](https://math.stackexchange.com/questions/1853774/determining-a-function-is-harmonic-from-mean-value-property-for-just-three-ra)) but in retrospect it might be more appropriate for this site.


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



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



Determining a function is harmonic from mean value property for just three(?) radii
Yes, in fact you only need two radii. More precisely, a theorem on page 167 of [this Monthly paper of Zalcman](http://dx.doi.org/10.2307/2321600) 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."


