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

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.