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The \(C^m\) norm of a function with prescribed jets. I. (English) Zbl 1228.42021

Let \(\vec P= \{P_x\}_{x\in S}\) be a family of polynomials on \(\mathbb{R}^n\) of degree \(k\) indexed by points of a finite set \(S\subset\mathbb{R}^n\). Define a norm of \(\vec P\) by
\[ \|\vec P\|:= \text{inf}\{\| F\|_{C^m}: \{T_x(F)\}_{x\in S}=\vec P\}, \]
where \(T_x(F)\) stands for the Taylor polynomial of \(F\) at \(x\) of degree \(k\). The author’s goal is to obtain an extension of the Whitney extension theorem
\[ \|\vec P\|\leq C(k,n)\max\{\|\vec P|_\Sigma\|: \Sigma\subset S,\;\operatorname{card}\Sigma\leq 2\}, \]
with \(C(k,n)\) growing exponentially in \(n^k\).
The author achieves this result with \(C(k,n)\) replaced by \((1+\varepsilon)\), \(\varepsilon> 0\) arbitrary, by exploiting the families \(\Sigma\subset S\) of cardinality at most \(\mu(\varepsilon)= \mu(\varepsilon, k,n)\); unfortunately, \(\mu(\varepsilon)\) is also enormously large.
Another version of this result uses a subfamily the above family of \(\Sigma\) satisfying the condition \(\| x-y\|\geq C(\varepsilon)\operatorname{diam}\Sigma\) for all \(x\neq y\) from \(\Sigma\).
Finally, the author proves an analogous result for the \(C^m\) extension of a function \(f: S\to\mathbb{R}\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
49K27 Optimality conditions for problems in abstract spaces
52A35 Helly-type theorems and geometric transversal theory
46E15 Banach spaces of continuous, differentiable or analytic functions
58C25 Differentiable maps on manifolds
26E10 \(C^\infty\)-functions, quasi-analytic functions
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References:

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