Monomial basis in Korenblum type spaces of analytic functions

Abstract

It is shown that the monomials $\Lambda =(z^n{)}_{n=0}^{\infty }$ are a Schauder basis of the Fréchet spaces $A_{+}^{-\gamma },\gamma \ge 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|{)}^{\mu }|f(z)|$ is bounded for all $\mu >\gamma$. Lusky \cite{L} 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.