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

## TL;DR

This paper analyzes Daubechies' time-frequency localization operator with a Gaussian window on Cantor-type fractal sets, providing explicit eigenvalue formulas and asymptotic estimates for the operator norm.

## Contribution

It introduces a framework for studying the operator on fractal sets, deriving explicit eigenvalues and asymptotic bounds for the operator norm on Cantor sets.

## Key findings

- Eigenvalues are explicitly computed for the Gaussian window case.
- Asymptotic estimates for the operator norm on Cantor sets are established.
- The results connect fractal geometry with time-frequency analysis.

## 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.