Extension of range of functions

By BY E. J. McSHANE, E. J. McShane. 1934

A well known and important theorem of analysis states that a function f(x) which is continuous on a bounded closed set E can be extended to the entire space, preserving its continuity. Let us consider a metric space S and a function f(x) defined and possessing a property P on a subset E of S. We shall for the sake of brevity say that ƒ (x) can be extended to S preserving property P , if there exists a function 4>(x), defined and possessing property P on all of S, which is equal to f(x) for all x on E. Our present object is to establish an easily proved theorem which both includes the classical theorem stated above, and also shows that functions satisfying a Lipschitz or Holder condition on an arbitrary set E can be extended to 5 preserving the Lipschitz or Holder condition. An advantage of the present procedure is that it yields an explicit formula for the extension, f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

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).”
On integral probability metrics, φ-divergences and binary classification (2009, background): “We quote the following result on Lipschitz extensions from [34] (see also [59], [60]).”
Homomorphisms on Function Lattices (2004, background): “9 and the fact that every Lipschitz function f : hðKÞ ! R possesses a Lipschitz extension to X (see McShane [22]).”
Embedding Problems in Symplectic Geometry (2005, background): “12 (Mc Shane [27]!) Consider a subset W of the metric space (X, d) and a function f : W —> R which is X-Lipschitz continuous.”
ABSOLUTELY MINIMIZING LIPSCHITZ EXTENSIONS ON A METRIC SPACE (2002, background): “Note that the set of supersolutions is never empty because the upper McShane– Whitney extension is a supersolution.”
Tug-of-war and the infinity Laplacian (2006, background): “The largest and smallest minimal extensions (introduced by McShane [18] and Whitney [24] in the 1930’s) are respectively”
Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time (2009, background): “Pioneer works in the extension of Lipschitz functions on general sets are [46, 38, 66, 67] but the interested reader can also consult [31] for an updated survey on the topic.”
The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values (2006, background): “Now p can be extended by the McShane extension, see [19], to R so that p and p do not change and (3.”
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).”
4. Monotone Dynamical Systems (2005, methodology): “According to McShane [134], a Lipschitz vector field on an arbitrary subset of Hv can be extended to a Lipschitz vector field on all of Hv, preserving the Lipschitz constant.”
On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls (2019, unknown): “ = op(1)fromLemma A.1. Therefore, |QJ(µ)−QJ(µ′)| ≤ K˜ kµ− µ′k, where K˜ = O p(1) and does not depend on the parameter µ. We deﬁne K used in function He J(µ) as the Lipschitz constant from QJ(µ). From McShane (1934), it follows that HeLB J (µ) is Lipschitz with constant K. Moreover, we know that K ≤ ”
The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces (2010, unknown): “By McShane’s extension theorem there exists a √ Nλ-Lipschitz extension ψ : l∞ (X)→ RN of φ ◦ ι−1 : A→ RN [25].”
Rigidity for Quasi-Möbius Group Actions (2000, methodology): “ a map h′ : Z→ Sk which is essential, constant on Z−B(p,r), and L r -Lipschitz where Lis independent of pand r. To obtain h′, we choose a>0, take a maximal ar-separated subset M⊂ B(p,r), and apply [11] to extend the restriction of hto M∪ (Z− B(p,r)) to a L1 r -Lipschitz map h1: Z→ Rk+1, where L1 ”