Monomial basis in Korenblum type spaces of analytic functions

TL;DR

This paper demonstrates that monomials form a Schauder basis in certain Korenblum type spaces of analytic functions, providing a sequence space representation and exploring related (LB)-spaces.

Contribution

It establishes the Schauder basis property of monomials in Korenblum type spaces and offers a new sequence space representation, extending previous results.

Findings

Monomials form a Schauder basis in $A_+^{-\gamma}$ spaces.

Sequence space representation of $A_+^{-\gamma}$ is provided.

Analysis of (LB)-spaces $A_{-}^{-\gamma}$ is included.

Abstract

It is shown that the monomials $\Lambda=(z^n)_{n=0}^{\infty}$ are a Schauder basis of the Fr\'echet spaces $A_+^{-\gamma}, \ \gamma \geq 0,$ 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 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.