C²interpolation with range restriction

Given-∞< λ <Λ<∞, E ⊂Rn finite, and f:E→ [λ,Λ], how can we extend f to a Cm(Rn) function F such that λ ≤ F ≤ Λ and ||F|| Cm(Rn) is within a constant multiple of the least possible, with the constant depending only on m and n? In this paper, we provide the solution to the problem for the case m = 2. Specifically, we construct a (parameter-dependent, nonlinear) C2(Rn) extension operator that preserves the range [λ,Λ], and we provide an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E).