Interpolation problems for functions with zero integrals over balls of fixed radius

Let $V_r(\mathbb{R}^n)$, $n\ge2$, be the set of functions $f\in L_{\operatorname{loc}}(\mathbb{R}^n)$ 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, the multiple interpolation problem is solved under general assumptions. For problems with an infinite set of nodes, sufficient solvability conditions are founded. Additionally, we construct a new example of a subset in $\mathbb{R}^n$ for which some nontrivial real analytic function of the class $V_r(\mathbb{R}^n)$ vanishes.