A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit $N$-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin's results obtained for the two-dimensional case by methods of function theory are transferred to the $N$-dimensional case ($N\ge3$). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.