The polynomial approximation of the hypograph of a function can be recast as a two-sided space-time L-1 minimization problem, related with the Lasso problem. In this paper, we solve this problem within an optimal control framework, which in turn provides insights to develop efficient computation algorithms. We prove existence and uniqueness of the optimal solution and we characterize it by means of the Pontryagin maximum principle. We establish convergence properties as the polynomial degree tends to +infinity. We provide numerical simulations to illustrate our results. In passing, we study the geometry and, in particular, the extremal points of the convex set of polynomials of one variable having two-sided constraints on an interval.