The "Three-Square#' Theorem for Continuous Functions

Thus for "most" choices of three squares /]1, P2, P3 the validity of (1) for -{P~, P2, P3} implies f ---0. The Pompeiu problem has an extensive history which we will not discuss here. The reader is referred to the paper [13] of L. ZALCMAN for an excellent account of the problem, including the "two-circles theorem" of Delsarte for harmonic functions, which is the motivation for our Theorem 1. We do mention, however, the following equivalent formulation of the Pompeiu problem, which is known as the Morera problem: Suppose all the sets P of the family ~ have "nice" boundaries (say piecewise smooth, so that the Green theorem can be applied to change integrations from P to OP). When is it true that the only continuous functions f such that