Some results of topological genericity

TL;DR

This paper demonstrates topological genericity results for various function spaces on the unit disc, showing that typical functions have unbounded Taylor coefficients and derivatives, with implications for harmonic and Hardy spaces.

Contribution

It establishes new topological genericity results for functions in Hardy, harmonic, and localized spaces, highlighting typical unbounded coefficient behavior.

Findings

Generic functions in these spaces have unbounded Taylor coefficients.

Most functions' derivatives also exhibit unboundedness.

Harmonic conjugates generally do not belong to any h^q space for q > 0.

Abstract

We show topological genericity for the set of functions in the space X, where X denotes the intersection of the Hardy spaces H^p with p<1, on the open unit disc such that the sequence of Taylor coefficients of the function and of all derivatives of the function are unbounded. Results of similar nature are valid when the space X is replaced by H^p(0 < p < 1) and by localized versions of such spaces. Looking at the smaller space A(D) \subseteq H^{\infty} we show topological genericity for the set of functions in A(D) and of all derivatives such that the sequence of Taylor coefficients of the function are outside of (\el)^1. We also show topological genericity for the set of functions in the space Y, where Y denotes the intersection of the harmonic Hardy spaces h^p with p<1, whose harmonic conjugate does not belong in any h^q (q > 0)