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Interpolation and extrapolation of smooth functions by linear operators. (English) Zbl 1084.58003

Let \( m \) and \( n \) be two integers with \( m\geq 1 \), \( n\geq 1 \). Denote by \( C^{m-1,1}({\mathbb R}^n) \) the Banach space of \( (m-1) \)-times differentiable functions in \( {\mathbb R}^n \) whose \( (m-1)^{th} \) derivatives are Lipschitz. Given a subset \( E \) of \( {\mathbb R}^n \) and a function \( \sigma : E\rightarrow [0,+\infty[ \), define \( C^{m-1,1}(E,\sigma) \) as the space of functions \( f :E\rightarrow {\mathbb R} \) for which there exist a function \( F \in C^{m-1,1}({\mathbb R}^n) \) and a constant \( M>0 \) such that \( \| F\|_{C^{m-1,1}({\mathbb R}^n)}\leq M \) and \( | F(x)-f(x)| \leq M\sigma (x) \) for all \( x\in E \). This is a Banach space with the norm \( \| f \|_{C^{m-1,1}(E,\sigma)} \) defined as the infimum of all such \( M \).
The author shows that there exists a linear map \( T: C^{m-1,1}(E,\sigma)\rightarrow C^{m-1,1}({\mathbb R}^n) \) and a constant \( A \) depending only on \( m \) and \( n \) such that, for any \( f\in C^{m-1,1}(E,\sigma) \) with \( \| f \|_{C^{m-1,1}(E,\sigma)}\leq 1 \), we have \( \| Tf\|_{C^{m-1,1}({\mathbb R}^n)}\leq A \) and \( | Tf(x)-f(x)| \leq A\sigma(x) \) for all \( x\in E \). In particular, taking \( \sigma=0 \) on \( E \), we get a continuous linear extension operator from the space \( C^{m-1,1}(E) \) of restrictions to \( E \) of elements of \( C^{m-1,1}({\mathbb R}^n) \), to the space \( C^{m-1,1}({\mathbb R}^n) \). The general result is obtained as the limit case (in some suitable sense) of an interpolation theorem for finite sets, with suitable continuity properties. The proofs rely heavily on refinements of the paper [Ann. Math. (2) 161, No. 1, 509–577 (2005; Zbl 1102.58005)] by the same author. The reader is assumed to be thoroughly familiar with that paper.

MSC:

58C25 Differentiable maps on manifolds
26E10 \(C^\infty\)-functions, quasi-analytic functions
52A35 Helly-type theorems and geometric transversal theory

Citations:

Zbl 1102.58005
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References:

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