Nearly optimal meshes in subanalytic sets
It is shown that nearly optimal meshes can be constructed on fat, compact, subanalytic subsets of \(\mathbb{R}^n\) that are known to admit certain Markov inequality [\textit{W. Pawłucki} and \textit{W. Pleśniak}, Math. Ann. 275, 467--480 (1986; Zbl 0579.32020)]. More precisely, the authors prove the following. Let \( K\) be a fat, compact, subanalytic subset of \(\mathbb{R}^n\). Then one can construct an admissible mesh \((A(d))\) on \(K\) such that \(\#A(d) = O((d \ln d)^N)\) as \(d\) tends to \(\infty\). In general, this provides better estimates of the cardinality of accessible meshes in subanalytic sets than those obtained by \textit{J.-P. Calvi} and \textit{N. Levenberg} [J. Approx. Theory 152, No. 1, 82--100 (2008; Zbl 1145.41001), Theorem 5].
scientific article; zbMATH DE number 6101121