Analyzing the Weyl-Heisenberg Frame Identity
Abstract
In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg 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
to 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; (5) In general, there are WH-frames and functions
f∈
L
2
(R)
so that the series on the right does not converge (even symmetrically). We give necessary and sufficient conditions for it to converge symmetrically; (6) There are WH-frames for which the series on the right always converges symmetrically to give the WH-Frame Identity, but there are functions for which the series does not converge and we classify when the series converges for all functions
f∈Ł
; (7) There are WH-frames for which the series always converges, but it does not converge unconditionally for some functions, and we classify when we have unconditional convergence for all functions f; and (8) We show that the series converges unconditionally for all
f∈
L
2
(R)
if g satisfies the CC-condition.