Monomial basis in Korenblum type spaces of analytic functions

Abstract

It is shown that the monomials Λ = ( z n ) n = 0 ∞ \Lambda =(z^n)_{n=0}^{\infty } are a Schauder basis of the Fréchet spaces A + − γ , γ ≥ 0 , A_+^{-\gamma }, \ \gamma \geq 0, that consists of all the analytic functions f f on the unit disc such that ( 1 − | z | ) μ | f ( z ) | (1-|z|)^{\mu }|f(z)| is bounded for all μ > γ \mu > \gamma . Lusky proved that Λ \Lambda is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type H ∞ H^{\infty } . A sequence space representation of the Fréchet space A + − γ A_+^{-\gamma } is presented. The case of (LB)-spaces A − − γ , γ > 0 , A_{-}^{-\gamma }, \ \gamma > 0, that are defined as unions of weighted Banach spaces is also studied.