MONOMIAL CONVERGENCE ON l(r)

Abstract

[EN] We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over l(r) for 1 < r <= 2. For H-b. (l(r)), the space of entire functions of bounded type in l(r), we prove that mon H-b (l(r)) is exactly the Marcinkiewicz sequence space m(psi r), where the symbol psi(r) is given by psi(r) (n) :=log(n + 1)(1-1/r) for n is an element of N-0.

For the space of m-homogeneous polynomials on l(r), we prove that the set of monomial convergence mon P((m)l(r)) contains the sequence space l(q), where q = (mr')'. Moreover, we show that for any q <= s < infinity, the Lorentz sequence space l(q),(s) lies in mon P((m)l(r)), provided that m is large enough. We apply our results to make an advance in the description of the set of monomial convergence of H-infinity (B-lr) (the space of bounded holomorphic functions on the unit ball of l(r)). As a byproduct we close the gap on certain estimates related to the mixed unconditionality constant for spaces of polynomials over classical sequence spaces.