In this paper, the author presents an interpolation space approach to the new area of quantitative approximation theory, namely, to approximation with constraints, which might be useful in many applications for the multivariate case. The author chooses a relatively simple, highly nontrivial approximation model, and demonstrates how the results of interpolation theory for operators preserves a convex cone structure. In particular, the author proves that the corresponding approximation result is related to the intersection property of the cone of nonnegative functions, with respect to the couple (L_p, B_p^{&agr; ∞} ).