Traces of functions of Zygmund class

Shvartsman, Pavel

doi:10.1007/bf00969335

Cited by 35 publications

(67 citation statements)

References 2 publications

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“…Since ξ → F β,ξ is linear for |β| ≤ m, we may drop the assumption |ξ| ≤ 1 from (19). Next, we return to (3), and conclude that, for |β| ≤ m and |ξ| ≤ 1, we have…”

Section: ⎤ ⎦mentioning

confidence: 90%

Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

Fefferman¹

2009

Rev. Mat. Iberoam.

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“…Since ξ → F β,ξ is linear for |β| ≤ m, we may drop the assumption |ξ| ≤ 1 from (19). Next, we return to (3), and conclude that, for |β| ≤ m and |ξ| ≤ 1, we have…”

Section: ⎤ ⎦mentioning

confidence: 90%

Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

Fefferman¹

2009

Rev. Mat. Iberoam.

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“…Evidently, the U ν = Ω form an open cover of Ω; and δ(x) > 0 on Ω. For each , ν, we have F ν , χ ν ∈ C m (Ω), thanks to (36) and (24). Also, hypotheses (GPU1,2,3,5,6,7) are immediate from our results (25), (26), (27), (29), (36), (48), respectively.…”

Section: Analysis On R N Ementioning

confidence: 91%

“…The general case was proven in Fefferman [10]. See Brudnyi-Shvartsman [ [27,28,29], Glaeser [21], Shvartsman [23,24,25], and Zobin [30,31] for several related results and conjectures. The proof of Theorem 2 is again constructive.…”

Section: Problemmentioning

confidence: 99%

The $C^m$ Norm of a Function with Prescribed Jets I

Fefferman¹

2010

Rev. Mat. Iberoam.

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We prove a variant of the classical Whitney extension theorem, in which the C m -norm of the extending function is controlled up to a given, small percentage error. IntroductionHere and in [16], we compute the least possible (infimum) C m norm of a function F having prescribed Taylor polynomials at N given points of R n . Moreover, given > 0, we exhibit such an F, whose C m norm is within of the least possible. Our computation consists of an algorithm, to be implemented on an (idealized) digital computer. The algorithm works, thanks to a variant of the classical Whitney extension theorem, in which we control the C m norm of the extending function up to an percentage error. This paper gives the variant of Whitney's theorem, while [16] presents the algorithm and the rest of the mathematics behind it. The number of operations used by our algorithm is C( )N log N, where N is the number of given points, and C( ) grows rapidly as tends to zero.To state our results precisely, we set up notation. Fix m, n ≥ 1. We pick a norm on C m (R n ), subject to restrictions to be spelled out in the next section. We write F C m (R n ) to denote the norm of F. Given x ∈ R n and F ∈ C m (R n ), we write J x (F) to denote the m th order Taylor polynomial of F at x. Thus, J x (F) belongs to P, the vector space of all (real) m th degree polynomials on R n . Let E ⊂ R n . We write #(E) for the number of points in E. (If E is infinite, then #(E) = +∞.) A Whitney field on E is a family P = (P x ) x∈E of polynomials P x ∈ P, indexed by x ∈ E. If P = (P x ) x∈E is a Whitney field and S ⊂ E is a subset, then in an obvious way we can define the restriction P| S of P to S.

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“…As is shown in [Shv2], , the statement of Theorem 1.1 (for the case of the family A k (R n )) is a geometrical background of so-called "finiteness property" of traces of smooth functions. For instance, for the Sobolev space W 2 ∞ (R n ) of functions whose distributional derivatives of the second order belong to L ∞ (R n ) this property looks as follows:…”

Section: P Shvartsman Gafamentioning

confidence: 97%