W. Pleśniak

Nearly optimal meshes in subanalytic sets

We prove that any fat, subanalytic compact subset of

A local version of Levenberg's theorem on determining measures and Leja's polynomial condition

\mathbb R^N

Approximation and Extension of C∞ Functions Defined on Compact Subsets of ℂn

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

Polynomial condition of Leja

\mathbb R^N

Extension of

onto a Markov compact set K in

C^{∞}

\mathbb C^l

functions from sets with polynomial cusps

(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

Quasianalytic functions in the sense of Bernstein

\mathbb R^N

A Cantor regular set which does not have Markov's property

. Versions for

Siciak's extremal function in complex and real analysis

\mathcal C^k