2021/07/17 by Charles Fefferman, Fefferman, Charles, Fushuai Jiang +3
Computer Science · Mathematics · #26B05 #41A05 #Classical Analysis and ODEs (math.CA) #Cryptography and Residue Arithmetic #FOS: Mathematics #Numerical Methods and Algorithms #Optimization and Control (math.OC) #Polynomial and algebraic computation #math.CA #math.OC #msc:26B05 #msc:41A05
paper · pdf · doi:10.48550/arxiv.2107.08272
62 pages. arXiv admin note: text overlap with arXiv:2102.05777
arxiv created 2021/07/17 · openalex publication_date 2021/07/17 · arxiv updated 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(Nlog N) operations, where N = #(E) .