J. Korean Math. Soc. 2004; 41(1): 209-229
Printed January 1, 2004
Copyright © The Korean Mathematical Society.
Andreas Defant, Domingo Garcia, and Manuel Maestre
Fachbereich Mathematik, Universitaet, Universidad de Valencia, Universidad de Valencia
Let $\sum_{|\alpha|=m} s_{\alpha}z^\alpha$, $z \in \mathbb C^n$ be a unimodular $m$-homoge-neous polynomial in $n$ variables (i.e., $|s_\alpha|=1$ for all multi indices $\alpha$), and let $R \subset \mathbb C^n$ be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modulus $\sup_{z\in R}| \sum_{|\alpha|=m}{\hskip-0.05cm} s_{\alpha}z^\alpha|$, and upper estimates for the average of these maximum moduli taken over all possible $m$-homogeneous Bernoulli polynomials (i.e., $s_\alpha=\pm 1$ for all multi indices $\alpha$). Examples show that for a fixed degree $m$ our estimates, for rather large classes of domains $R$, are asymptotically optimal in the dimension $n$.
Keywords: several complex variables, power series, polynomials, Banach spaces, unconditional basis, Banach-Mazur distance
MSC numbers: Primary 32A05; Secondary 46B07, 46B09, 46G20
2004; 41(1): 77-94
2018; 55(3): 695-704
2016; 53(6): 1391-1409
2012; 49(6): 1139-1161
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd