$C^2$ interpolation with range restriction

Abstract

Given −∞ < λ < Λ < ∞, E ⊂ R finite, and f : E → [λ,Λ], how can we extend f to a C(R) 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) C(R) 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).