• DOI:10.1006/ACHA.2001.0374
• Corpus ID: 119709007

Analyzing the Weyl–Heisenberg Frame Identity

@article{Casazza1999AnalyzingTW,
  title={Analyzing the Weyl–Heisenberg Frame Identity},
  author={Peter G. Casazza and Mark Lammers},
  journal={Applied and Computational Harmonic Analysis},
  year={1999},
  volume={12},
  pages={171-178}
}

• P. Casazza, M. Lammers
• Published 4 November 1999
• Mathematics
• Applied and Computational Harmonic Analysis

Abstract In 1990, Daubechies proved a fundamental identity for Weyl–Heisenberg systems which is now called the Weyl–Heisenberg (WH)-frame identity. WH-frame identity: If g ∈ W ( L ∞ , L 1 ), then for all continuous, compactly supported functions f we have ∑ m , n ∣〈f,E mb T na g〉∣ 2 = 1 b ∑ k ∫ R f ( t ) f(t−k/b) ∑ n g(t−na) g ( t − na − k / b ) dt. It has been folklore that the identity will not hold universally. We make a detailed study of the WH-frame identity and show (1) The identity does not require any assumptions on ab (such as the requirement that ab ≤1 have a frame); (2) As stated above, the identity holds for all f ∈ L 2 ( R ); (3) The identity holds for all bounded, compactly supported functions if and only if g ∈ L 2 ( R ); (4) The identity holds for all compactly supported functions if and only if ∑ n ∣ g ( x − na )∣ 2 ≤ B a.e. Moreover, in (2)–(4) above, the series on the right converges unconditionally. We will also see that in general, symmetric, norm, and unconditional convergences of the series in the WH-frame identity are all different.