Analyzing the Weyl–Heisenberg Frame Identity

✍ Scribed by Peter G Casazza; Mark C Lammers

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 91 KB
• Volume: 12
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.2001.0374

✦ Synopsis

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

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(xna)| 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.