Quantum theta functions and Gabor frames for modulation spaces

TL;DR

This paper explores the deep connections between quantum theta functions from noncommutative geometry and Gabor frames in signal analysis, revealing shared algebraic structures and functional equations.

Contribution

It establishes a bridge between quantum theta functions and Gabor analysis, showing their underlying algebraic similarities and how they inform each other.

Findings

Quantum theta functions arise naturally from Gabor frame representations.

Rieffel scalar product underpins functional equations for quantum thetas.

Fundamental Identity of Gabor analysis relates to associativity in noncommutative geometry.

Abstract

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.