A Hilbert Lemniscate Theorem in ℂ 2

Thomas Bloom[1]; Norman Levenberg[2]; Yu. Lyubarskii[3]

  • [1] University of Toronto Toronto (Canada)
  • [2] Indiana University Bloomington, IN 47405 (USA)
  • [3] Norwegian University of Science and Technology Trondheim, 7491 (Norway)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 6, page 2191-2220
  • ISSN: 0373-0956

Abstract

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For a regular, compact, polynomially convex circled set K in C 2 , we construct a sequence of pairs { P n , Q n } of homogeneous polynomials in two variables with deg P n = deg Q n = n such that the sets K n : = { ( z , w ) ∈ C 2 : | P n ( z , w ) | ≤ 1 , | Q n ( z , w ) | ≤ 1 } approximate K and if K is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set { P n = Q n = 1 } 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.

How to cite

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Bloom, Thomas, Levenberg, Norman, and Lyubarskii, Yu.. "A Hilbert Lemniscate Theorem in $\mathbb{C}^2$." Annales de l’institut Fourier 58.6 (2008): 2191-2220. <http://eudml.org/doc/10375>.

@article{Bloom2008,
abstract = {For a regular, compact, polynomially convex circled set $K$ in $\mathbf\{C\}^2$, we construct a sequence of pairs $\lbrace P_n,Q_n\rbrace $ of homogeneous polynomials in two variables with $\{\rm deg\}\,P_n = $$\{\rm deg\}\,Q_n =n$ such that the sets $K_n: = \lbrace (z,w) \in \mathbf\{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.},
affiliation = {University of Toronto Toronto (Canada); Indiana University Bloomington, IN 47405 (USA); Norwegian University of Science and Technology Trondheim, 7491 (Norway)},
author = {Bloom, Thomas, Levenberg, Norman, Lyubarskii, Yu.},
journal = {Annales de l’institut Fourier},
keywords = {Logarithmic potential; Monge-Ampère measure; subharmonic functions; atomization; logarithmic potential},
language = {eng},
number = {6},
pages = {2191-2220},
publisher = {Association des Annales de l’institut Fourier},
title = {A Hilbert Lemniscate Theorem in $\mathbb\{C\}^2$},
url = {http://eudml.org/doc/10375},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Bloom, Thomas
AU - Levenberg, Norman
AU - Lyubarskii, Yu.
TI - A Hilbert Lemniscate Theorem in $\mathbb{C}^2$
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 2191
EP - 2220
AB - For a regular, compact, polynomially convex circled set $K$ in $\mathbf{C}^2$, we construct a sequence of pairs $\lbrace P_n,Q_n\rbrace $ of homogeneous polynomials in two variables with ${\rm deg}\,P_n = $${\rm deg}\,Q_n =n$ such that the sets $K_n: = \lbrace (z,w) \in \mathbf{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.
LA - eng
KW - Logarithmic potential; Monge-Ampère measure; subharmonic functions; atomization; logarithmic potential
UR - http://eudml.org/doc/10375
ER -

References

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