Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions

Brudnyi, Yuri; Shvartsman, Pavel

doi:10.1090/s0002-9947-01-02756-8

Cited by 62 publications

(54 citation statements)

References 7 publications

(1 reference statement)

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“…Here, we will only mention that the results of this nature for the special class of convex functions have interesting applications in differential geometry, PDE theory (such as Monge–Ampère equations), non‐linear dynamics and quantum computing (see and the references therein). We should also note here that, in contrast with the classical Whitney extension theorem (concerning jets) and also with the solutions to the Whitney extension problem (concerning functions), in whose proofs one can use appropriate partitions of unity in order to patch local solutions together to obtain a global solution, such tools are no longer available in our setting. Moreover, further difficulties arise from the rigid global behaviour of convex functions (see Proposition , for instance).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, none of these

F

can be convex on

double-struckR

, because, as is easily checked, any convex extension

g

f

double-struckR

must satisfy

g (x) = | x |

for every

x \in [a, d]

, and therefore

g

cannot be differentiable at 0. On the other hand, it should be noted that following the works of Brudnyi–Shvartsman for the solution of the

C^{1, 1}

Whitney extension problem, and of Fefferman concerning the solutions of the

C^{m - 1, 1}

and

C^{m}

Whitney extension problems in full generality, there is a natural interpretation of the term local condition in extension theorems that refers to the existence of a finiteness principle , which states that extendibility (with controlled norm) of a function from all finite subsets of

C

of cardinality at most

k

(for some

k < \infty

fixed) implies extendibility (with controlled norm) of the function defined on all of

C

. It would be very interesting to know whether such a finiteness principle holds for

C^{1, 1}

convex extension of functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Whitney extension theorems for convex functions of the classes C1 and C1,ω

Azagra

Mudarra

2017

Proc. London Math. Soc.

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Let C be a subset of R n (not necessarily convex), f : C → R be a function and G : C → R n be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f , G for the existence of a convex function F ∈ C 1,ω (R n ) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C 1 convex functions on R n , with a good control of the Lipschitz constants of the extensions (namely, Lip(F ) G ∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of R n by boundaries of C 1 or C 1,1 convex bodies with prescribed outer normals on K.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, none of these

F

can be convex on

double-struckR

, because, as is easily checked, any convex extension

g

f

double-struckR

must satisfy

g (x) = | x |

for every

x \in [a, d]

, and therefore

g

cannot be differentiable at 0. On the other hand, it should be noted that following the works of Brudnyi–Shvartsman for the solution of the

C^{1, 1}

Whitney extension problem, and of Fefferman concerning the solutions of the

C^{m - 1, 1}

and

C^{m}

C

of cardinality at most

k

(for some

k < \infty

fixed) implies extendibility (with controlled norm) of the function defined on all of

C

. It would be very interesting to know whether such a finiteness principle holds for

C^{1, 1}

convex extension of functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Whitney extension theorems for convex functions of the classes C1 and C1,ω

Azagra

Mudarra

2017

Proc. London Math. Soc.

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“…A much more difficult problem is the Whitney problem [34] of extending functions to C 1,1 or C m,1 functions on R n . It is a topic of extensive research; see [13,14,16,8].…”

Section: Introductionmentioning

confidence: 99%

Continuity of extensions of Lipschitz maps

Ciosmak

2021

Isr. J. Math.

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We establish the sharp rate of continuity of extensions of R m -valued 1-Lipschitz maps from a subset A of R n to a 1-Lipschitz maps on R n . We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that if m > 1, then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of R n . This shows a striking difference from the case m = 1, where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map v is when the difference between the two maps takes values in a fixed one-dimensional subspace of R m and the set A is geodesically convex with respect to a Riemannian pseudo-metric associated with v.

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“…The classical Whitney Interpolation Problem 3 is well-understood thanks to the works of Brudnyi and Shvartsman [7,9,10], Fefferman and Klartag [14,15,18,21,22]. In [21,22], the authors provide an efficient algorithm for solving the classical Whitney Interpolation Problem 3.…”

Section: Introductionmentioning

confidence: 99%

“…Problem 4 and the related "Finiteness Principles" (see e.g. Theorem 1.4 below) have been extensively studied by Y. Brudnyi and P. Shvartsman [6,10], C. Fefferman, A. Israel, and G.K. Luli [20], C. Fefferman and P. Shvartsman [24].…”

Section: Introductionmentioning

confidence: 99%

$C^2$ Interpolation with Range Restriction

Fefferman¹,

Jiang²,

Luli³

2021

Preprint

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) 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) C 2 (R n ) extension operator that preserves the range [λ, Λ], and we provide an efficient algorithm to compute such an extension using O(N log N) operations, where N = #(E).

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Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions

Cited by 62 publications

References 7 publications

Whitney extension theorems for convex functions of the classes C1 and C1,ω

Whitney extension theorems for convex functions of the classes C1 and C1,ω

Continuity of extensions of Lipschitz maps

$C^2$ Interpolation with Range Restriction

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