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Monomial basis in Korenblum type spaces of analytic functions


Authors: José Bonet, Wolfgang Lusky and Jari Taskinen
Journal: Proc. Amer. Math. Soc. 146 (2018), 5269-5278
MSC (2010): Primary 46E10; Secondary 46A35, 46A45, 46E15
DOI: https://doi.org/10.1090/proc/14195
Published electronically: September 17, 2018
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Abstract: It is shown that the monomials $ \Lambda =(z^n)_{n=0}^{\infty }$ are a Schauder basis of the Fréchet spaces $ A_+^{-\gamma }, \ \gamma \geq 0,$ that consists of all the analytic functions $ f$ on the unit disc such that $ (1-\vert z\vert)^{\mu }\vert f(z)\vert$ is bounded for all $ \mu > \gamma $. Lusky proved that $ \Lambda $ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $ H^{\infty }$. A sequence space representation of the Fréchet space $ A_+^{-\gamma }$ is presented. The case of (LB)-spaces $ A_{-}^{-\gamma }, \ \gamma > 0,$ that are defined as unions of weighted Banach spaces is also studied.


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Additional Information

José Bonet
Affiliation: Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain
Email: jbonet@mat.upv.es

Wolfgang Lusky
Affiliation: Institut für Mathematik, Universität Paderborn, D-33098 Paderborn, Germany
Email: lusky@uni-paderborn.de

Jari Taskinen
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki, 00014 Helsinki, Finland
Email: jari.taskinen@helsinki.fi

DOI: https://doi.org/10.1090/proc/14195
Received by editor(s): January 17, 2018
Received by editor(s) in revised form: April 16, 2018
Published electronically: September 17, 2018
Additional Notes: Research of the first author was partially supported by the project MTM2016-76647-P.
Research of the third author was partially supported by the Väisälä Foundation of the Finnish Academy of Sciences and Letters.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2018 American Mathematical Society

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