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Bernstein–Walsh inequalities and the exponential curve in $\mathbb {C}^2$


Authors: Dan Coman and Evgeny A. Poletsky
Journal: Proc. Amer. Math. Soc. 131 (2003), 879-887
MSC (2000): Primary 41A17; Secondary 30D15, 30D20
DOI: https://doi.org/10.1090/S0002-9939-02-06571-1
Published electronically: June 12, 2002
MathSciNet review: 1937426
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for the pluripolar set $K=\{(z,e^z): |z|\leq 1\}$ in ${\Bbb C}^2$ there is a global Bernstein–Walsh inequality: If $P$ is a polynomial of degree $n$ on ${\Bbb C}^2$ and $|P|\leq 1$ on $K$, this inequality gives an upper bound for $|P(z,w)|$ which grows like $\exp (\frac 12n^2\log n)$. The result is used to obtain sharp estimates for $|P(z,e^z)|$.


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Additional Information

Dan Coman
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150
MR Author ID: 325057
Email: dcoman@syr.edu

Evgeny A. Poletsky
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150
MR Author ID: 197859
Email: eapolets@syr.edu

Received by editor(s): June 8, 2001
Received by editor(s) in revised form: October 18, 2001
Published electronically: June 12, 2002
Additional Notes: The second author was partially supported by NSF Grant DMS-9804755
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2002 American Mathematical Society