Quantum Theta Functions and Gabor Frames for Modulation Spaces

Open Access
Article

DOI: 10.1007/s11005-009-0306-7

Cite this article as:
luef, F. & Manin, Y.I. Lett Math Phys (2009) 88: 131. doi:10.1007/s11005-009-0306-7

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 will 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.

Mathematics Subject Classification (2000)

Primary 42C15  46L89 Secondary 11F27 14K25 

Keywords

quantum tori Gabor frames Weil representation theta functions 
Download to read the full article text

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.University of ViennaViennaAustria
  2. 2.University of CaliforniaBerkeleyUSA
  3. 3.Max-Planck-Institut für MathematikBonnGermany
  4. 4.Northwestern UniversityEvanstonUSA

Personalised recommendations