Cantor型集上的Daubechies时频局部化算子II
Daubechies' Time-Frequency Localization Operator on Cantor Type Sets II
摘要：
我们研究了联合时频表示中的分形不确定性原理。也就是说，我们考虑Daubechies的局部化算子投影到具有任意基$M>1$和字母表$\mathscr{A}$的球对称$n$-Cantor集上。我们推导出一个上界渐近线的乘法常数的运营商规范的基础$M$和字母大小$|\mathscr{A}|$的康托尔集。对于任何固定的基础和字母表的大小，我们表明，有康托集，使渐近线是最佳的。特别地，渐近线对于第一部分中研究的中三分之一康托尔集是精确的。尽管如此，这并不延伸到每一个康托尔集，因为我们提供的例子中，最佳渐近线没有实现。
英文摘要：
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.
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