Daubechies' Time-Frequency Localization Operator on Cantor Type Sets

Abstract

We study Daubechies' time-frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time-frequency representation, we define the $n$-iterate spherically symmetric Cantor set in the joint representation. For the $n$-iterate Cantor set, precise asymptotic estimates for the operator norm are then derived up to a multiplicative constant.