Monomial expansions on sequence spaces

MURO, SANTIAGO

Madrid

Conferencia; Function Theory on Infinite Dimensional Spaces XVI.; 2019

Universidad Complutense de Madrid

    For each entire function $f$ on $n$ complex variables, there is a series of monomials, the monomial expansion of $f$, such that         $$f(z)=sum_{alpha}a_alpha z^alpha,$$        for every $z$ and the convergence is uniform on each compact set.        If $f$ is a holomorphic function on an infinite dimensional sequence space $X$, then it also has a monomial expansion, but in this case, the series does not necessarily converge for every $zin X$.                 There has been some effort to characterize the subset of $X$ where the monomial expansion of every holomorphic function on $mathcal F$ converge, where $mathcal F$ is a family of polynomials or of holomorphic functions on $c_0$ or $ell_p$. This set is called the {it set of monomial convergence} of $mathcal F$. The only case where set of monomial convergence has been completely characterized is when $X=ell_1$, or $mathcal F=mathcal P (^m c_0)$.                In this talk we will describe        the set of monomial convergence for the space $H_b(ell_r)$ of entire functions of bounded type on $ell_r$, for $1 < r leq 2$, and show that it is exactly a Marcinkiewicz sequence space.                 We will also talk about the set of monomial convergence for $mathcal P (^m ell_r)$, the space of $m$-homogeneous polynomials on $ell_r$,        and for the space $H^infty(B_{ell_r})$ of bounded holomorphic on the unit ball of $ell_r$, $1 < r leq 2$.                The talk is based on a joint work with Daniel Galicer (Universidad de Buenos Aires), Mart´in Mansilla (Universidad de Buenos Aires) and Pablo Sevilla-Peris (Universidad Polit´ecnica de Valencia).