C² interpolation with range restriction

Given -∞< λ < Λ < ∞ , E ⊂ R^n finite, and f E → [λ,Λ] , how can we extend f to a C^m(R^n) function F such that λ≤ F ≤ Λ and \|F\|C^m(R^n) 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) C^2(R^n) extension operator that preserves the range [λ,Λ] , and we provide an efficient algorithm to compute such an extension using O(Nlog N) operations, where N = #(E) .