Monomial basis in Korenblum type spaces of analytic functions

It is shown that the monomials $\mathrm{\Lambda }=(z^n{)}_{n=0}^{\mathrm{\infty }}$ are a Schauder basis of the Fr'echet spaces $A_{+}^{-\gamma },\gamma \ge 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-\mathrm{\mid }z\mathrm{\mid }{)}^{\mu }\mathrm{\mid }f(z)\mathrm{\mid }$ is bounded for all $\mu >\gamma$. Lusky \cite{L} proved that $\mathrm{\Lambda }$ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $H^{\mathrm{\infty }}$. A sequence space representation of the Fr'echet 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.