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- W2552354516 abstract "The Bernstein Markov Property for a compact set E and a positive finite mea-sure μ supported on E is a strong comparability assumption between L μ 2 and uni-form norms on E of polynomials (or other nested families of functions) as theirdegree tends to infinity.Admissible meshes are sequences of sampling sets A k ⊂ E whose cardinalityis growing sub-exponentially with respect to k and for which there exists a positivefinite constant C such that max E |p| ≤ C max A k |p| for any polynomial of degree atmost k.These two mathematical objects have several applications and motivations fromApproximation Theory and Pluripotential Theory, the study of plurisubharmonicfunctions in several complex variables.The properties of Bernstein Markov measures and admissible meshes for agiven compact set E are very similar, indeed they may be seen as the continuousand the discrete approach to the same problem.This work is concerned on providing sufficient conditions for some differentinstances of the Bernstein Markov property and explicitly constructing admissiblemeshes.As first problem, we study sufficient conditions for a version of the BernsteinMarkov property for rational functions on the complex plane and its relation withthe polynomial Bernstein Markov property.In Chapter 5, we consider the case of a compact subset E of an algebraic purem-dimensional subset A of C n and we prove a sufficient condition for the BernsteinMarkov property for the traces of polynomials on E.To this aim, we provide two new results in Pluripotential Theory regarding theconvergence and the comparability of the relative capacities, the relative and globalextremal functions and the Chebyshev constants on a (possibly non-smooth) purem-dimensional algebraic variety in C n , which are of independent interest.In the last part of the dissertation, we provide some construction proceduresfor admissible meshes on some classes of real compact sets.Finally, we present some algorithms, based on admissible meshes, for thenumerical approximation of the most relevant objects in Pluripotential Theory,namely, the transfinite diameter, the Siciak Zaharjuta extremal function and thepluripotential equilibrium measure." @default.
- W2552354516 created "2016-11-30" @default.
- W2552354516 creator A5079852837 @default.
- W2552354516 date "2016-01-30" @default.
- W2552354516 modified "2023-09-26" @default.
- W2552354516 title "Bernstein Markov Properties and Applications" @default.
- W2552354516 hasPublicationYear "2016" @default.
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