Time-frequency analysis on the Heisenberg group

Abstract

It is the main goal of this text to study certain aspects of time-frequency analysis on the 2n+1-dimensional Heisenberg group. More specifically, we will discuss how the well-studied notions of modulation spaces and Weyl quantization can be extended from the Euclidean space Rn to the Heisenberg group Hn.

For quite a long time already this group has served as a good test object to verify which concepts and results from Euclidean (thus Abelian) analysis carry over to simple instances of non-Abelian structures.

In the case of the Weyl quantization a reasonable answer for $\H$ was first proposed by A. S. Dynin almost forty years ago, although it was studied in more detail only some twenty years after that by G. B. Folland. We will review the foundations laid by Dynin and Folland and present some new results about left-invariant differential operators and the natural product of symbols, the Moyal product.

The special tool for our analysis is a $3$-step nilpotent Lie group to which we will refer as the Dynin-Folland group. As the name suggests it originates in the works of the afore-mentioned authors. The group's unitary irreducible representations are in fact the key to both the Weyl quantization and modulation spaces on Hn.

Our results on modulation space on the Heisenberg group are based on H. Feichtinger and K. Grochenig's coorbit theory and a more recent adaption of it by I. and D. Beltictua, which focuses on modulation spaces arising from nilpotent Lie groups. We will use a blend of both approaches and discuss the modulation spaces induced by the Dynin-Folland group, among them a type of modulation spaces on Hn.