A constructive proof of a Whitney extension theorem in one variable

Archer, J. C.; Gruyer, Erwan Le

doi:10.1016/0021-9045(92)90122-5

Cited by 5 publications

(9 citation statements)

References 8 publications

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“…( 1) |a − a | ≥ 3 for every a, a ∈ A, a a ; Of course, the first inequality in (5.72) is immediate from ( 1). Prove the second inequality.…”

Section: Extension Criteria For Sobolev W M P -Functionsmentioning

confidence: 98%

“…Here ν = 1 or 2. For instance, we can produce the families T (1) j and T (2) j by (i) enumerating the intervals from T j in "increasing order" (recall that these intervals are disjoint) and (ii) setting T (1) j to be the family of intervals from T j with the odd index, and T (2) j to be the family of intervals from T j with the even index.…”

Section: Extension Criteria For Sobolev W M P -Functionsmentioning

confidence: 99%

“…provided I ν ∅; otherwise, we set B (ν) k = 0. Clearly, B k = B (1) k + B (2) k + B (3) k . Prove that…”

Section: The Variational Criterion For W Mmentioning

confidence: 99%

“…Without loss of generality, we may assume that I ν ∅ for each ν = 1, 2, 3. First we estimate B (1) k . We note that i + k − m ≥ 0 for each i ∈ I 1 ; otherwise x i+k−m = −∞ (see Convention 5.3) which contradicts the inequality x i+k − x i+k−m ≤ 2.…”

Section: The Variational Criterion For W Mmentioning

confidence: 99%

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Sobolev functions on closed subsets of the real line

Shvartsman

2020

Journal of Approximation Theory

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For each p > 1 and each positive integer m we give intrinsic characterizations of the restriction of the Sobolev space W m p (R) and homogeneous Sobolev space L m p (R) to an arbitrary closed subset E of the real line.In particular, we show that the classical one dimensional Whitney extension operator [67] is "universal" for the scale of L m p (R) spaces in the following sense: for every p ∈ (1, ∞] it provides almost optimal L m p -extensions of functions defined on E. The operator norm of this extension operator is bounded by a constant depending only on m. This enables us to prove several constructive W m p -and L m p -extension criteria expressed in terms of m th order divided differences of functions.Integrating this inequality on J with respect to y, we obtain that |J| |∆ k G[S ]| q ≤ 2 q |J| q J |G (k+1) (x)| q dx + 2 q J |G (k) (y)| q dy.

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“…( 1) |a − a | ≥ 3 for every a, a ∈ A, a a ; Of course, the first inequality in (5.72) is immediate from ( 1). Prove the second inequality.…”

Section: Extension Criteria For Sobolev W M P -Functionsmentioning

confidence: 98%

Section: Extension Criteria For Sobolev W M P -Functionsmentioning

confidence: 99%

“…provided I ν ∅; otherwise, we set B (ν) k = 0. Clearly, B k = B (1) k + B (2) k + B (3) k . Prove that…”

Section: The Variational Criterion For W Mmentioning

confidence: 99%

Section: The Variational Criterion For W Mmentioning

confidence: 99%

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Sobolev functions on closed subsets of the real line

Shvartsman

2020

Journal of Approximation Theory

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“…The sufficiency of (W n ) can be deduced from [9, theorem 3.1.14]. 1 However, this and other proofs of Whitney's Extension Theorem (see, e.g., [1,2,10]) are quite long, mainly due to their generality and framework considered (of dealing with partial functions on R n ). Thus, for a sake of simplicity and completeness of this article, we include a simple proof of this theorem, in our particular form, in Section 7.…”

Section: Whitney's Extension Theorem and Other Preliminariesmentioning

confidence: 99%

Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom

Ciesielski¹,

Seoane‐Sepúlveda²

2018

Real Analysis Exchange

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The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν ∈ ω ∪ {∞} there exists a family Fν ⊂ C ν (R) of cardinality ω1 < c such that for every g ∈ D ν (R) the set g \ Fν has cardinality ≤ ω1. Moreover, we show that this result remains true for partial functions g (i.e., g ∈ D ν (X) for some X ⊂ R) if, and only if, ν ∈ {0, 1}. The proof of this result is based on the following theorem of independent interest (which, for ν = 0, seems to have been previously unnoticed): for every X ⊂ R with no isolated points, every ν-times differentiable function g : X → R admits a ν-times differentiable extensionḡ : B → R, where B ⊃ X is a Borel subset of R. The presented arguments rely heavily on a Whitney's Extension Theorem for the functions defined on perfect subsets of R, which short but fully detailed proof is included. Some open questions are also posed.

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Smooth extension theorems for one variable maps

Ciesielski

2019

Journal of Mathematical Analysis and Applications

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A constructive proof of a Whitney extension theorem in one variable

Cited by 5 publications

References 8 publications

Sobolev functions on closed subsets of the real line

Sobolev functions on closed subsets of the real line

Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom

Smooth extension theorems for one variable maps

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