Analytic Extensions of Differentiable Functions Defined in Closed Sets

By H. Whitney. 1934

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Citations in the corpus

On the empirical estimation of integral probability metrics (2012, methodology): “Hence, by Proposition 11.2.3 of [15], gα in (2.5) extends any g defined on {X1, . . . , XN} (with g(Xi) = a⋆i and ‖g‖BL = 1) to S, i.e., gα(Xi) = g(Xi), ∀ i and ‖gα‖BL = ‖g‖BL. Note that hα in (2.6) is the Lipschitz extension of f to S (by McShane-Whitney Lipschitz extension theorem).”
Breaking the Curse of Dimensionality with Convex Neural Networks (2014, background): “For α > 1, we have the upper-bound E sup γ1(f)6δ |J(f)− Ĵ(f)| 6 2Gδ√ n + 2GδαE sup ‖v‖p61/R ∣ ∣ ∣ ∣ 1 n n ∑ i=1 τiv ⊤zi ∣ ∣ ∣ ∣ using the α-Lipschitz-cont. of (·)α+ on [−1, 1], 6 2 Gδ√ n + 2 Gαδ Rn E ∥ ∥ ∥ ∥ n ∑ i=1 τizi ∥ ∥ ∥ ∥ q .”
Geometry-Informed Neural Operator for Large-Scale 3D PDEs (2023, background): “Continuity of g can be ensured by an appropriate extension for any x ∈ D such that dist ( x, ∂D ) ϵ and dist ( x, ∂D ) ≥ ϵ/ 2 [17].”
Linear programming based Lyapunov function computation for differential inclusions (2011, background): “Indeed, since V (x(t)) is monotone decreasing and bounded from below by 0 we obtain V (x(t)) ↘ c∗ ≥ 0.”
Interpolation and extrapolation of smooth functions by linear operators (2005, background): “By the standard Whitney extension theorem (see [11] or [15]), (10.18) and (10.19), there exists a function F̃ ∈ Cm(Rn), with (10.21) ‖F̃‖Cm(Rn) ≤ C , and (10.22) Jxµ(F̃ ) = Pµ(f, y) for eachµ = 1, . . .”
Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan (2018, unknown): “Theorem 46 (Whitney (1934), [168]).”
Normal linear stability of quasi-periodic tori (2007, methodology): “Finally, by the Whitney Extension Theorem we can extend the limit Φ∞ to a smooth map Φ defined on Tn × Rn × R2q × U such that Φ satisfies conclusions of Theorem 1.6.”
A review of linear response theory for general differentiable dynamical systems (2009, background): “How then could the function k → ρk be differentiable? One idea [40] is to use differentiability in the sense of Whitney [ 50 ]: find a differentiable function φ such that φ(k) = ρk(A) when k belongs to some set � ⊂ S and define the derivative dρk(A)/dk to be φ ′ (k) when k ∈ �.”
Conformal prediction with conditional guarantees (2023, unknown): “These proxies can always be extended to a complete function on all of X using the methods of McShane (1934), Whitney (1934).”
THE BOUNDEDNESS OF CLASSICAL OPERATORS ON VARIABLE L p SPACES (2006, methodology): “[45] Whitney, H.: Analytic extensions of differentiable functions defined in closed sets.”
Sobolev extension by linear operators (2012, background): “, m = 1) holds with T given by the classical proof of the Whitney extension theorem [25].”
Uniqueness of ∞-Harmonic Functions and the Eikonal Equation (2007, background): “…= v V ⇒ Lip u V ≤ Lip v V (1.2) alternatively, since there always exists v satisfying Lip v V = Lip u V this amounts to Lip u V = Lip u V for every V U The special v’s just mentioned are known as the McShane–Whitney extensions (McShane, 1934; Whitney, 1934, or, for example, Aronsson et al., 2004).”
A Hölder infinity Laplacian (2012, background): “Indeed, we can take one of the extensions from [14] and [21]”
Reeb Spaces of Smooth Functions on Manifolds (2020, background): “Then, by [30], there always exists a smooth function f : M → [0,∞) such that f(0) is a Cantor set embedded in M .”