Abstract
We prove that any fat, subanalytic compact subset of \(\mathbb R^N\) possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389–398, 2010).) We also show that an infinitely differentiable map f from a compact set Q in \(\mathbb R^N\) onto a Markov compact set K in \(\mathbb C^l\) (l ≤ N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389–398, 2010) for analytic maps in case Q is a subset of \(\mathbb R^N\). Versions for \(\mathcal C^k\) maps with sufficiently large k are also given.
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Pleśniak, W. Nearly optimal meshes in subanalytic sets. Numer Algor 60, 545–553 (2012). https://doi.org/10.1007/s11075-012-9578-6
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DOI: https://doi.org/10.1007/s11075-012-9578-6
Keywords
- Admissible polynomial meshes
- Optimal meshes
- Subanalytic geometry
- Hironaka rectilinearization theorem
- Bernstein-Walsh-Siciak theorem
- Jackson theorem