Pluripotential Numerics

Abstract

We study the numerical approximation of the fundamental quantities in pluripotential theory, namely the Siciak Zaharjuta extremal plurisubharmonic function $$V_E^*$$ of a compact $$\mathcal {L}$$ -regular set $$E\subset {\mathbb {C}}^n$$ , its transfinite diameter $$\delta (E),$$ and the pluripotential equilibrium measure $$\mu _E:=\left( {{\mathrm{dd^c}}}V_E^*\right) ^n$$ . 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 $$\delta (E)$$ and the local uniform convergence of our approximation to $$V_E^*$$ on $${\mathbb {C}}^n$$ . Then the convergence of the proposed approximation of $$\mu _E$$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with $$E\subset \mathbb {R}^2$$ show the performance of the proposed methods.