On the singular factor of a linear combination of holomorphic functions

TL;DR

This paper investigates the singular inner factors of linear combinations of bounded holomorphic functions, showing they are generally limited under certain smoothness conditions but not in all cases.

Contribution

It establishes conditions under which linear combinations of bounded holomorphic functions have few singular inner factors, highlighting the role of boundary smoothness.

Findings

Linear combinations have few singular inner factors if functions are smooth up to the boundary.

Without smoothness assumptions, the number of singular inner factors can be large.

Smoothness up to the boundary is crucial for controlling singular inner factors.

Abstract

We prove that the linear combinations of functions $f_0,...,f_n$ in $H^\infty$ have "few" singular inner factors, provided that the $f_j$'s are suitably smooth up to the boundary, while in general this is no longer true.