Wiener's Tauberian theorem in classical and quantum harmonic analysis

TL;DR

This paper extends Wiener's Tauberian theorem to quantum harmonic analysis, introduces new versions, characterizes slowly oscillating operators, and explores their relation to compact operators and applications in operator theory.

Contribution

It presents new versions of Wiener's Tauberian theorem, formulates operator analogues in quantum harmonic analysis, and characterizes a broader class of slowly oscillating operators.

Findings

New versions of Wiener's Tauberian theorem for limit functions

Characterization of slowly oscillating operators as larger than compact operators

Application to operator theory and uniform versions of the theorem

Abstract

We investigate Wiener's Tauberian theorem from the perspective of limit functions, which results in several new versions of the Tauberian theorem. Based on this, we formulate and prove analogous Tauberian theorems for operators in the sense of quantum harmonic analysis. Using these results, we characterize the class of slowly oscillating operators and show that this class is strictly larger than the class of compact operators. Finally, we discuss uniform versions of Wiener's Tauberian theorem and its operator analogue and provide an application of this in operator theory.