Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

TL;DR

This paper investigates the boundary behavior of Taylor series in Hardy and Bergman spaces, showing that universality occurs on maximal exceptional sets and establishing the mixing property of the Taylor backward shift in certain Bergman spaces.

Contribution

It demonstrates the sharpness of boundary universality results for Hardy and Bergman spaces and proves the mixing property of the Taylor backward shift in these contexts.

Findings

Universality occurs on maximal exceptional sets in Hardy and Bergman spaces.

The Taylor backward shift is mixing on certain Bergman spaces.

Results are sharp, indicating boundary behavior differs from classical cases.

Abstract

It is known that, generically, Taylor series of functions holomorphic in the unit disc turn out to be universal series outside of the unit disc and in particular on the unit circle. Due to classical and recent results on the boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman spaces the situation is essentially different. In this paper it is shown that in many respects these results are sharp in the sense that universality generically appears on maximal exceptional sets. As a main tool it is proved that the Taylor (backward) shift on certain Bergman spaces is mixing.