Seminar talk – Erling Svela

On March 24, 2025, at 3 PM in the Buzano Room, we will have the pleasure of hosting a talk by

Erling Svela (NTNU)

who is visiting our group for an extended period. He will present a seminar titled

Daubechies’ theorem for mixed-state localization operators.

Abstract. Let \( f, g \in L^2(\mathbb{R}^d) \), and \( \Omega \subset \mathbb{R}^{2d} \) be measurable. We define the (pure state) localization operator with mask \( \chi_\Omega \) and window \( g \) by:

\[
A^g_\Omega f = \int_\Omega \langle f, \pi(z) g \rangle \pi(z) g \, dz,
\]

where \( \pi(z) g(t) = \pi(x, \omega) g(t) = e^{2\pi i \langle t, \omega \rangle} g(t – x) \). Localization operators originate in time-frequency analysis, but also play an important role in quantum harmonic analysis. They are the simplest examples of function-operator convolution, namely between the function \( \chi_\Omega \) and the rank-one operator \( g \otimes g \). We can therefore view the general function-operator convolution as:

\[
\chi_\Omega \ast S(f) = \int_\Omega \pi(z) S \pi(z)^* f \, dz
\]

as a “mixed-state” localization operator.

A fundamental result in the theory of localization operators is Daubechies’ theorem, which gives a complete spectral description of a localization operator when \( \Omega \) is a disc and \( g \) is the Gaussian. In this talk, we extend Daubechies’ theorem to multivariate mixed-state localization operators. If \( \Omega \) is a Reinhardt domain and \( S \) is a polyradial operator, we show that a similar spectral description holds for the mixed-state localization operator \( \chi_\Omega \ast S \). We discuss elements of the result’s proof, and applications to time-frequency concentration problems.