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Zbl 1152.32015
Bloom, Thomas; Levenberg, Norman; Lyubarskii, Yu.
A Hilbert lemniscate theorem in $\Bbb C^{ 2 }$. (English)
[J] Ann. Inst. Fourier 58, No. 6, 2191-2220 (2008). ISSN 0373-0956; ISSN 1777-5310/e

Summary: For a regular, compact, polynomially convex circled set $K$ in $\Bbb C^{ 2 }$, we construct a sequence of pairs $\lbrace P_{n},Q_{ n }\rbrace $ of homogeneous polynomials in two variables with $\deg P_{n}=\deg Q_{n}=n$ such that the sets $K_{ n }:=\lbrace (z,w)\in \Bbb C^{ 2 }:|P_{ n }(z,w)|\le 1,|Q_{ n }(z,w)|\le 1\rbrace $ approximate $K$ and if $K$ is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set $\lbrace P_{ n }=Q_{ n }=1\rbrace $ converge to the pluripotential-theoretic Monge-Ampère measure for $K$. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.

MSC 2000:: *32U05 Plurisubharmonic functions and generalizations
32W20 Complex Monge-Ampère operators
31C05 Generalizations of harmonic (etc.) functions
31C15 Generalizations of potentials, etc.

Keywords: logarithmic potential; Monge-Ampère measure; subharmonic functions; atomization

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