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from Part 3 - Replacing Polydiscs by Other Balls
Published online by Cambridge University Press: 19 July 2019
Given a function f on the n-dimensional polydisc, the Bohr radius (recall Chapter 8) looks for the best r for which the supremum of ∑ | c_α z^α| for || z ||_∞ <r is less than or equal to the supremum of |f(z)| for || z ||_∞ <1. Here an analogous problem is considered, replacing the sup-norm by another p-norm. The corresponding Bohr radius for l_p-balls is defined, and its asymptotic behaviour is computed. This is done in three steps. First, an m-homogeneous version (where only m-homogeneous polynomials are considered) is defined, and it is shown how these m-homogeneous radii determine the general Bohr radius. In the second step, this homogenous radius is related to the unconditional basis constant of the monomials in the space of homogeneous polynomials on l_p. Finally, this unconditional basis constant is computed.
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