The Pompeiu problem revisited

Abstract

One of the central results connected with the Pompeiu problem is a theorem of Brown, Schreiber and Taylor. Using some old work of the authors on spectral synthesis, a proof of this result is given. Though separately dealt with, it is shown that some of the main results for the Pompeiu problem for non-Euclidean symmetric spaces can also be treated in the same sprit. In all the cases the role of representations of the underlying group of isometries is highlighted. This point of view leads to some new results for the Pompeiu problem for two sided translations on the non-commutative grouups SL(2, R) and M(2). Finally, a brief discussion is provided for some related problems.