## 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

### Keywords:

Whitney extension theorem; optimal $$C^m$$-norm; $$k$$-jet
Full Text:

### References:

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