Resumen de C 2 interpolation with range restriction

Charles L. Fefferman , Fushuai Jiang, Garving K. Luli

• Given −∞<<Λ<∞−∞<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 2m=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(NlogN) operations, where N=#(E) .