\(C^{m,\omega}\) extension by bounded-depth linear operators
Let \(S\subset\mathbb{R}^n\) be a closed set and \(C^{k,\omega}[S]\) be the trace of the Banach space \(C^{k,\omega}(\mathbb{R}^n)\) to \(S\). It was proved by \textit{H.\,Whitney} [Trans.\ Am.\ Math.\ Soc.\ 36, 63--89 (1934; JFM 60.0217.01 and Zbl 0008.24902)] for \(n= 1\), \(k\geq 1\), \textit{Yu.\,A.\thinspace Brudnyi} and \textit{P.\,Shvartsman} [J.~Geom.\ Anal.\ 7, No.\,4, 515--574 (1997; Zbl 0937.58007)] for \(n\geq 1\), \(k= 1\) and \textit{Ch.\,Fefferman} [Rev.\ Mat.\ Iberoam.\ 25, No.\,1, 1--48 (2009; Zbl 1173.46014)] for the general case that there exists a linear extension operator \(L: C^{k,\omega}[S]\to C^{k,\omega}(\mathbb{R}^n)\) whose norm is bounded by a constant depending only on \(k\) and \(n\). The operators constructed in the first two cited papers are well-localized in the sense that for all \(x\in \mathbb{R}^n\) and \(f\in C^{k,\omega}[S]\) there exist points \(x^i\in S\) and numbers \(\lambda_i\), \(1\leq i\leq d\), such that \[ (Lf)(x)= \sum^\infty_{i=1} \lambda_i f(x^i), \] where the \(x_i,\lambda_i\) depend only on \(x\), while \(d\) depends only on \(k\) and \(n\). The minimal number \(d\) is called depth of \(L\). As for the Fefferman construction, it was proved that the operator \(L\) has finite depth depending only on \(k\) and \(n\) if the set \(S\) is finite, see \textit{Ch.\,Fefferman} [loc.\,cit.]. Adding some new ingredients to the Fefferman construction and following the main line of his proof, the author constructs a linear extension operator \(L: C^{k,\omega}[S]\to C^{k,\omega}(\mathbb{R}^n)\) of finite depth \(d= d(k,n)\) for any closed subset \(S\). Moreover, he proves some additional properties of \(L\) that may be seen as the Whitney divided difference construction. For instance, for every non-limit point \(x\) of \(S\), there exists a subset \(\Sigma_x\subset S\) of cardinality at most \(d(k,n)\) depending only on \(x\) such that the Taylor polynomial \(T^k_x(Lf)\) is uniquely and linearly defined by \(f/\Sigma_x\).
scientific article; zbMATH DE number 5729703
scientific article