Badly Approximable Unimodular Functions in Weighte

Authors: Alexei B Aleksandrov

Publish Date: 2013/03/07

Volume: 4, Issue: 2, Pages: 315-326

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Abstract

A function u on the unit circle T is said to be badly approximable in the weighted space L pTw if ¦¦u + f¦¦ L pTw ≥¦L pTw for all f ∊ H∞ We prove that if an unimodular function u is badly approximable in L pT w for all p ∊ 0 +∞ and some nonzero weight w then overlineu is an inner function We describe the inner functions Θ and the weights w on the unit circle T such that Θ is badly approximable in L pT w for all p 0 It turns out that for given inner functions Θ the class of all weights satisfying the abovementioned condition depends only on the zero set of Θ In other words Θ is badly approximable in L pT w for all p ∊ 0 +∞ if and only if overlineB is badly approximable in L pTw for all p ∊ 0 +∞ where B is a Blaschke product with simple zeros and such that Θ−10 = B−10

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