Daubechies' Time-Frequency Localization Operator on Cantor Type Sets II

TL;DR

This paper investigates the behavior of Daubechies' localization operator on Cantor sets within the time-frequency domain, establishing bounds for the operator norm and identifying cases of optimal asymptotic behavior.

Contribution

It derives an upper bound for the operator norm on Cantor sets and shows the existence of sets where this bound is sharp, especially for the mid-third Cantor set.

Findings

Upper bound for the operator norm in terms of base and alphabet size

Existence of Cantor sets with optimal asymptote

Mid-third Cantor set achieves the precise asymptote

Abstract

We study a version of the fractal uncertainty principle in the joint time-frequency representation. Namely, we consider Daubechies' localization operator projecting onto spherically symmetric $n$-iterate Cantor sets with an arbitrary base $M>1$ and alphabet $\mathscr{A}$. We derive an upper bound asymptote up to a multiplicative constant for the operator norm in terms of the base $M$ and alphabet size $|\mathscr{A}|$ of the Cantor set. For any fixed base and alphabet size, we show that there are Cantor sets such that the asymptote is optimal. In particular, the asymptote is precise for the mid-third Cantor set, which was studied in part I. Nonetheless, this does not extend to every Cantor set as we provide examples where the optimal asymptote is not achieved.