Interpolation problems for functions with zero ball means

Let $n\geq 2$, $V_r(\mathbb{R}^n)$ be the set of functions with zero integrals over all balls in $\mathbb{R}^n$ of radius $r$. Various interpolation problems for the class $V_r(\mathbb{R}^n)$ are studied. In the case when the set of interpolation nodes is finite, we solve the interpolation problem under general conditions. For the problems with infinite set of nodes, some sufficient conditions of solvability are founded.
Note that an essential condition is that the definition of the class $V_r(\mathbb{R}^n)$ involves integration over balls. For instance, it can be shown that the analogues of our results in which the class of functions is defined using zero integrals over all shifts of a fixed parallelepiped in $\mathbb{R}^n$ do not hold true.