Quantum Harmonic Analysis on Locally Compact Abelian Groups

• Published in: The Journal of fourier analysis and applications Vol. 31; no. 1
• Main Authors: Fulsche, Robert, Galke, Niklas
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2025; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximation; Approximations and Expansions; Fourier Analysis; Group theory; Harmonic analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Operators; Partial Differential Equations; Signal,Image and Speech Processing; Theorems; 43A25; Quantum harmonic analysis; Locally compact abelian group; Operator convolution; Primary: 47B90; Secondary: 43A65
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-024-10140-9

We extend the notions of quantum harmonic analysis , as introduced in R. Werner’s paper from 1984 (J Math Phys 25(5):1404–1411), to abelian phase spaces, by which we mean a locally compact abelian group endowed with a Heisenberg multiplier. In this way, we obtain a joint harmonic analysis of functions and operators for each such phase space. Throughout, we spend significant extra effort to include also phase spaces which are not second countable. We obtain most results from Werner’s paper for these general phase spaces, up to Wiener’s approximation theorem for operators. In addition, we extend certain of those results (most notably Wiener’s approximation theorem) to operators acting on certain coorbit spaces affiliated with the phase space.