Pluripotential Numerics

We study the numerical approximation of the fundamental quantities in pluripotential theory , namely the Siciak Zaharjuta extremal plurisubharmonic function V E ∗ of a compact L -regular set E ⊂ C n , its transfinite diameter δ ( E ) , and the pluripotential equilibrium measure μ E : = dd c V E ∗ n...

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Bibliographic Details
Published in:Constructive approximation 2019-04, Vol.49 (2), p.227-263
Main Author: Piazzon, Federico
Format: Article
Language:English
Subjects:
Acquisitions & mergers
Algorithms
Analysis
Approximation
Convergence
Markov processes
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
Polynomials
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Summary:We study the numerical approximation of the fundamental quantities in pluripotential theory , namely the Siciak Zaharjuta extremal plurisubharmonic function V E ∗ of a compact L -regular set E ⊂ C n , its transfinite diameter δ ( E ) , and the pluripotential equilibrium measure μ E : = dd c V E ∗ n . The developed methods rely on the computation of a polynomial mesh for E ,  for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of δ ( E ) and the local uniform convergence of our approximation to V E ∗ on C n . Then the convergence of the proposed approximation of μ E follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with E ⊂ R 2 show the performance of the proposed methods.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-018-9441-7