Solution of the support problem for several function classes

Several classes of functions with zero integrals over all balls of a fixed radius are studied. A description of functions in such classes as series in special functions is obtained and a theorem on a removable singularity is proved. These results make it possible to solve completely the support problem for several classes of functions with zero ball averages. As a consequence, a far-reaching generalization of the well-known Zalcman's two-radii theorem is obtained.