10 - Monomial Convergence of Holomorphic Functions

Summary

Given a family of formal power series, its set of monomial convergence is defined as those z’s for which the series converges. The main focus is given to the sets of monomial convergence of the m-homogeneous polynomials on c0 and of the bounded holomorphic functions on B_{c0}. The first one is completely described in terms of the Marcinkiewicz space l_{(2m)/(m-1), ∞}. For the second one there is no complete description. If z is such that limsup (log n)^(1/2) ∑_j^n (z*_j)^{2} < 1 (where z* is the decreasing rearrangement of z), then z is in the set of monomial convergence of the bounded holomorphic functions. Also, if z belongs to the set of monomial convergence, then the limit superior is ≤ 1. This is related to Bohr’s problem (see Chapter 1). First of all, if M denotes the supremum over all q so that l_q is contained in the set of monomial convergence of the bounded holomorphic functions on Bc0, then S=1/M. But this can be more precise: S is the infimum over all σ >0 so that the sequence (p_n^(-σ))_n (being p_n the n-th prime number) belongs to the set of monomial convergence of the bounded holomorphic functions on Bc0.