Vol. 14, No. 3, 2021

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Monomial convergence on $\ell_r$

Daniel Galicer, Martín Mansilla, Santiago Muro and Pablo Sevilla-Peris

Vol. 14 (2021), No. 3, 945–984

We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over r for 1 < r 2. For Hb(r), the space of entire functions of bounded type in r, we prove that monHb(r) is exactly the Marcinkiewicz sequence space mΨr, where the symbol Ψr is given by Ψr(n) := log(n + 1)11r for n 0.

For the space of m-homogeneous polynomials on r, we prove that the set of monomial convergence mon𝒫(mr) contains the sequence space q, where q = (mr) . Moreover, we show that for any q s < , the Lorentz sequence space q,s lies in mon𝒫(mr), 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(Br) (the space of bounded holomorphic functions on the unit ball of 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.

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holomorphic function, homogeneous polynomial, monomial convergence, Banach sequence space
Mathematical Subject Classification 2010
Primary: 32A05, 46E50, 46G20, 46G25
Received: 12 July 2019
Accepted: 21 November 2019
Published: 18 May 2021
Daniel Galicer
Departamento de Matemática
Facultad de Cs. Exactas y Naturales
Universidad de Buenos Aires and IMAS-CONICET
Ciudad Universitaria
Buenos Aires
Martín Mansilla
Departamento de Matemática
Facultad de Cs. Exactas y Naturales
Universidad de Buenos Aires and IMAS-CONICET
Ciudad Universitaria
Buenos Aires
Santiago Muro
Facultad de Cs. Exactas, Ingenieria y Agrimensura
Universidad Nacional de Rosario and CIFASIS-CONICET
Ocampo y Esmeralda
Pablo Sevilla-Peris
Institut Universitari de Matemàtica Pura i Aplicada
Universitat Politècnica de València