Analytic extensions of differentiable functions defined in closed sets

1934/01/01 by Hassler Whitney · 862 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebra over a field #Differentiable function #Mathematics #Pure mathematics #Rough Sets and Fuzzy Logic #Submersion (mathematics)

paper · doi:10.1090/s0002-9947-1934-1501735-3

published in Transactions of the American Mathematical Society 36(1), 63-89 (American Mathematical Society)

openalex publication_date 1934/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A continuous extension the author has not seen in the literature may be given as follows; we assume for simplicity that A is bounded.Let h(r) (räO) be a continuous and monotone increasing function such that A(0)=0, and if x and y are any two points of A whose distance apart is rxv, then \f(x) -f(y) \úh(rxv).For any points x of E and y of A, set B(x, y)=f(y)-h(rxi); then if x is in .4,B(x,y)^f(x).The continuous extension of f(x) is,F(x), which at each point x of £ equals the maximum of H(x, y) as y varies over A. 63

Cited by

• The Modern Mathematics of Deep Learning
• From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations
• Dynamic models with p parameters are identified by 2p+1 random features
• On the classifying space of a Morse flow category
• Compact Embedding Theorem Associated with Classical Weight Functions in Two Variables
• Sharpness for C¹ linearization of planar hyperbolic diffeomorphisms
• Exponential Absolute Minimizing extension and biased infinity Laplacian
• Differentiable convex extensions with sharp Lipschitz constants
• Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization
• Generating all Ahlfors currents by a single entire curve
• Jump problem for generalized Lamé-Navier systems in ℝ^m
• On the differentiability of the Sobolev norm of W^k,1(M) on compact Riemannian manifolds
• Some results of the Lipschitz constant of 1-Field on ℝⁿ
• Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian
• Directional Subdifferentials at Infinity and Its Applications
• A novel approach through spherical functions in the characterization of invariant functions
• Neural Network Convergence for Variational Inequalities
• Continuous differentiability of the signum function and Newton's method for bang-bang control
• (m, ψ, δ)-capacity and its properties
• Posterior distribution of nondifferentiable functions
• A Mixed-Gauge Caratheodory Measure Bridging Lebesgue Volume and Surface Content
• The subcritical finite-volume massive sine-Gordon model
• A Nash-Kuiper theorem for isometric immersions in a high codimension
• Extending Bilipschitz Mappings between Separated Nets
• Global Lipschitz extension preserving the slope
• Linearizations and optimization problems in diffeological spaces
• Intrinsicality of second fundamentals of hypersurfaces in space form
• Computability of a Whitney Extension
• Feedback-driven recurrent quantum neural network universality
• C^1,1/3- very weak solutions to the two dimensional Monge–Ampère equation
• Structure of finite dimensional linear bundles morphisms
• Continuity and approximability of competitive spectral radii
• Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric
• Differentiable functions
• A sharp form of Whitney’s extension theorem
• Hilbert's Fifth Problem: Review
• ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem
• A lecture on the classical KAM theorem
• Axioms for Euclidean Green's functions II
• Axioms for Euclidean Green's functions
• Quelques propriétés globales des variétés différentiables
• Peak sets and boundary interpolation sets for the unit disc: a survey
• On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds
• Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation
• Whitney Approximation: domains and bounds
• Quasisymmetric rectifiability of uniformly disconnected sets
• The divergence of the BFGS and Gauss Newton methods
• BC-ADMM: An Efficient Non-convex Constrained Optimizer with Robotic Applications
• Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains. [europepmc]
• Fluctuations Around a Homogenised Semilinear Random PDE. [europepmc]
• Exception Sets of Intrinsic and Piecewise Lipschitz Functions. [europepmc]
• Absolutely minimal semi-Lipschitz extensions. [europepmc]

Related

• Differentiability versus continuity: Restriction and extension theorems and monstrous examples
• Restrictions of continuous functions
• Differentiable functions defined in closed sets. A problem of Whitney
• Non extendability from any side of the domain of definition as a generic property of smooth or simply continuous functions on an analytic curve
• Extending continuous functions