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