Interpolation of functions with zero spherical averages obeying growth constraints

Let $V_r({\Bbb R}^n)$, with $n\geq 2$ and $r>0$, be the set of locally integrable functions $f: {\Bbb R}^n\to {\Bbb C}$ with the zero integrals over all balls of radius $r$ in ${\Bbb R}^n$. We study the interpolation problem $f(a_k)=b_k$, with $k=1,2,\dots$, for functions in $(V_r\cap C^{\infty})({\Bbb R}^n)$ with growth constraints at infinity. Under consideration is the case that $\{a_k\}_{k=1}^{\infty}$ is a set of points on a certain straight line $l$ in ${\Bbb R}^n$ which is close in some sense to a finite union of arithmetic progressions and $\{b_k\}_{k=1}^{\infty}$ is a sequence of complex numbers satisfying the condition $\sum_{k=1}^{\infty}|b_k|^2<+\infty$. We show that this interpolation problem is solvable in the class of those functions in $(V_r\cap C^{\infty})({\Bbb R}^n)$ which, together with their derivatives, satisfy a special decay condition at infinity. The condition is an upper bound that implies power decay in the directions orthogonal to $l$ and also cannot be significantly improved along the straight line $l$.