Translation-Invariant Rotation-Invariant Spaces Of Continuous Functions And The Pompeiu Problem

Abstract:  . If V is a closed translation-invarian t rotation-invariant subspace of continuous functions on R 2 with the usual topology which contains nonpolynomial functions, then V contains a subspace of Helmholtz functions. In addition, a converse theorem is given showing when V is precisely a space of Helmholtz functions. We conclude with applications to a mean value theorem for the Helmholtz equation. 2 1 Introduction In 1929, the Roumanian mathematician Dimitrie Pompeiu formulated the following problem [11, 12]: Characterize the bounded measureable sets D ae R 2 which have the property that f j 0 is the only continuous function satisfying Z oe(D) f(x; y)dxdy = 0 for every oe 2 \Sigma (1) where \Sigma is the group of rigid motions in the plane. Such sets D are said to have the Pompeiu property. This general question (or variations thereof) is referred to as the Pompeiu problem. A large amount of research has gone into this problem (see Chamberland[6] and Zalcman[22] for surveys). Wh...