Generalized Carleson--Newman inner functions

TL;DR

This paper investigates a class of inner functions related to Banach algebras, introduces a singular function outside this class, and provides elementary proofs for existing results, advancing understanding of the weak embedding property.

Contribution

It constructs a singular inner function that cannot be embedded into the generalized Carleson-Newman class, answering an open question and simplifying some existing proofs.

Findings

Identified a singular function outside the class

Provided elementary proofs for previous results

Enhanced understanding of the weak embedding property

Abstract

We study a class of inner functions introduced by Gorkin, Mortini, and Nikolski, and motivated by Banach algebras and functional calculus applications. Answering their question, we produce a singular function that cannot be multiplied into this generalized Carleson-Newman class of inner functions (the class of functions satisfying the so called weak embedding property). Furthermore, we give elementary proofs for some results on these classes obtained earlier using arguments related to the Gelfand representation of $H^\infty$.