C 2 interpolation with range restriction

• Charles Fefferman ^[1] ; Fushuai Jiang ^[2] ; Garving K. Luli ^[3]
  1. [1] Princeton University

     Princeton University

     Estados Unidos

     
  2. [2] Fushuai Jiang
  3. [3] University of California at Davis, USA

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 2, 2023, págs. 649-710
• Idioma: inglés
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• Resumen

  • 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) .