Distance Formulas on Weighted Banach Spaces of Analytic Functions

Abstract

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function f 0 in the Banach space H ∞ v of all analytic functions f such that v|f | is bounded, the distance of f 0 to the subspace H 0 v of H ∞ v of all the functions g such that v|g| vanishes at infinity is attained at a function g 0 ∈ H 0 v . Moreover a simple, direct proof of the formula of the distance of f to H 0 v due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

“…A distance formula from an element f ∈ H ∞ v to the closed subspace H 0 v , which is the closure of the polynomials, was obtained by Perfekt [141,142]. A direct, elementary proof was presented in [57] and it is explained in Sect. 6.…”