Let \(E\) be a fat (i.e. \(E=\overline{\text{int}E}\)) compact subset of the space \(\mathbb R^N\). Let \(C^\infty_{\text{int}}(E)\) denote the space of all \(C^\infty\)functions in int\(E\) which can be continuously extended together with all their partial derivatives to \(E\). \(E\) is said to have Jackson’s property, if for each \(k=0,1, \dots\) there exist a positive constant \(C_k\) and a positive integer \(m_k\) such that for all \(f\in\; C^\infty_{\text{int}}(E)\) and all \(n>k\) one has \( n^k\text{dist}_E(f,\mathcal P_n)\leq C_k\| f\|_{E,m_k}\), where \(\| f\|_{E,m}:=\sum_{|\alpha|\leq m}\| D^{\alpha}f\|_E,\;m=0,1,2,\dots\) and \(\text{dist}_E(f,\mathcal P_n):=\inf\{\| f-p\|_E :p\in \mathcal P_n\}.\) Here \(\mathcal P_n\) is the space of all polynomials of degree at most \(n\) and \(\|\cdot\|_E\) denotes the Chebyshev norm. It is well-known that every compact cube in \(\mathbb R^N\) admits Jackson’s inequality (with \(m_k=k+1\)).
The purpose of this paper is to deliver other examples of Jackson sets. It is shown that a finite union of disjoint Jackson compact sets in \(\mathbb R^N\) is also a Jackson set and that this in general fails to hold for an infinite union of Jackson sets. In the class of fat compact sets \(E\) that admit Markov’s inequality (\(\|\text{gradient}\, p\|_E\leq M(\text{deg}\,p)^r\|p\|_E\) for every polynomial \(p\) with some positive constants \(M\) and \(r\) independent of \(p\)), \(E\) is a Jackson set if and only if it fulfils the following extension requirement: for each function \(f\in C^\infty_{\text{int}}(E)\), there is a function \(g\in C^\infty(\mathbb R^N)\) such that \(g_{|E}=f\). This together with a result of E. Bierstone [Bol. Soc. Bras. Mat. 11, No. 2, 139–189 (1980; Zbl 0584.58006)] permits one to show that Whitney regular compact subsets of \(\mathbb R^N\) are Jackson sets.