The Journal of Fourier Analysis and Applications
Volume 1, Number 4, 1995
Gabor Time-Frequency Lattices
and the Wexler–Raz Identity
Ingrid Daubechies, H. J. Landau, and Zeph Landau
ABSTRACT. Gabor time-frequency lattices are sets of functions of the form gmÞ;nþ .t/ =
e−2³ iÞmt g.t − nþ/ generated from a given function g.t/ by discrete translations in time and
frequency. They are potential tools for the decomposition and handling of signals that, like speech
or music, seem over short intervals to have well-defined frequencies that, however, change with
time. It was recently observed that the behavior of a lattice .mÞ; nþ/ can be connected to that of
a dual lattice .m=þ; n=Þ/. Here we establish this interesting relationship and study its properties.
We then clarify the results by applying the theory of von Neumann algebras. One outcome is a
simple proof that for gmÞ;nþ to span L2 , the lattice .mÞ; nþ/ must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having improved
convergence and localization properties.
1. Introduction
This paper concerns expansions of L2 .R/-functions f .x/ into families gmÞ;nþ .x/ obtained by
translating and modulating a fixed function g in L2 .R/,
gmÞ;nþ .x/ = e−2³ imÞx g.x − nþ/ ;
(1.1)
with Þ; þ > 0 fixed and m; n ranging over Z. We call such families Gabor time-frequency lattices
(or Gabor lattices, for short), after an expansion of this type proposed by Gabor in [2] (where g was
a Gaussian, and Þþ = 1). The general problem of identifying coefficients cm;n .f / so that
Math Subject Classification. 47C15, 94A11.
Keywords and Phrases. Wexler-Raz identity, time-frequency localization, Gabor expansions, frames, lattice density,
von Neumann algebra, coupling constant.
Acknowledgements and Notes. The first author thanks Nigel Lee for bringing the work of Wexler and Raz to her
attention. We thank A. J. E. M. Janssen as well as A. Ron and Z. Shen for sending us their then-unpublished
manuscripts [7] and [8]. The three approaches in [7], [8], and the present paper are completely different and were
developed independently and more or less simultaneously; all three papers also contain some results that do not appear
in the other two. Note, however, that Janssen’s paper (in this same journal) was submitted several months before ours,
so he should be given priority over us for the identity of the frame dual function and the WR dual function. We also
thank him for a very thorough reading of our manuscript, which led to a revision that improved the paper.
c
1995
CRC Press, Inc.
ISSN 1069-5869
438
I. Daubechies, H. J. Landau, and Z. Landau
f .x/ =
X
(1.2)
cm;n gmÞ;nþ .x/
m;n
has been discussed in many places. In [3] this problem was studied in the context of frames; the
gmÞ;nþ are said to constitute a frame if there exist A > 0 and B < ∞ such that, for all f ∈ L2 .R/,
A f 2 ≤
X
|f; gmÞ;nþ |2 ≤ B f 2 ;
(1.3)
m;n
where ; denotes the standard L2 -inner product, f; g =
The condition (1.3) can also be rewritten as
R
f .x/g.x/ dx, and f = f; f 1=2 .
A Id ≤ Sg;Þ;þ ≤ B Id ;
(1.4)
where Sg;Þ;þ is the frame operator, defined as
Sg;Þ;þ f =
X
f; gmÞ;nþ gmÞ;nþ :
(1.5)
m;n
If the gmÞ;nþ satisfy (1.3), then the general theory of frames [4] can be used to show that the least
squares choice for the cm;n in (1.2) is given by
−1
cm;n = f; Sg;Þ;þ
gmÞ;nþ = f; g̃ mÞ;nþ ;
(1.6)
−1
where the g̃ mÞ;nþ are derived from the single function g̃ = Sg;Þ;þ
g by the same time-frequency
−1
translations as in (1.1). Note that g̃ is well defined, since S is bounded by A−1 because of (1.4);
moreover, one can write a geometrically convergent series for S −1 because
S=
½
A+B
2S
;
Id − Id −
A+B
2
(1.7)
2S with Id − A+B
< 1. We will refer to g̃ as the frame dual function.
A different approach to finding appropriate cm;n was taken in [5]. By means of a remarkable
identity that we shall call the Wexler–Raz identity, the problem was linked to the dual Gabor lattice
gm=þ;n=Þ .x/. The authors of [5] show that if h satisfies
h; gm=þ;n=Þ = Þþ Žm;0 Žn;0 ;
(1.8)
then the coefficients cm;n = f; hmÞ;nþ will satisfy (1.2); their method for solving (1.8) involves
another operator, S , required to be bounded and invertible. Among all these different “biorthogonal”
choices h, they then choose the one with minimal L2 -norm; we will call it the Wexler–Raz dual
function and denote it by g # . (Strictly speaking, the construction in [5] is for `2 -sequences in
Gabor Time-Frequency Lattices
439
“discrete time” rather than for L2 -functions, and the derivation is not rigorous, but the essential idea
is as above. Janssen’s paper [6] gives a rigorous derivation of the Wexler–Raz result.)
As the functions g̃ and g # minimize different things (the norms of solutions of (1.2) and (1.8)
2
in ` and L2 , respectively), it is not immediately clear how they are related. In this paper, we show
in three different ways that they coincide. The most concrete of our proofs is mainly concerned with
the Wexler–Raz identity and establishes g̃ = g # as a byproduct. It occupies §§2 to 4 and is organized
as follows. In §2, we introduce our notation and review some properties of the “traditional” frame
construction. In §3, we state and prove the Wexler–Raz (WR) identity in operator form and explore
its consequences. In §4 we show that the WR approach to solving (1.8) can be carried out—that is,
S is bounded and invertible—if and only if the gmÞ;nþ constitute a frame and that one then always
has g # = g̃.
Our second proof sidesteps the Wexler–Raz identity and proves the result directly. This short
argument is given in §5.
Section 6 revisits the problem from the point of view of von Neumann algebras. The ad hoc
arguments of the first proof are seen to be concrete realizations of a much more elegant construction
using von Neumann algebras. This approach proves all the results that were obtained in the first part,
that is, the Wexler–Raz identity as well as the equivalence proved at the end of §4. Turning then to
the density of the lattice, we show by a simple argument that the condition Þþ ≤ 1 is necessary for
the functions {gmÞ;nþ } to span L2 . Heretofore this fact was proved by using the coupling constant, a
feature of von Neumann algebras that is difficult to establish. Here we reverse the direction of the
argument by using the Wexler–Raz identity to derive the existence of the coupling constant in an
elementary way.
Finally, part of the motivation of Wexler and Raz in [5] was that, in practice, the g # of minimal
2
L -norm may not be the preferred choice. They present several particular constructions for examples
where, for instance, a different “biorthogonal” function leads to better time concentration. We
conclude the present paper with a discussion in §7 of a more systematic approach to find biorthogonal
functions that optimize other than L2 -norms.
Some of our results overlap with those of [7] by A. J. E. M Janssen and [8] by A. Ron and
Z. Shen, of which we became aware while writing the present paper, but our techniques are different.
2. The Frame Dual Function — A Quick Review
In this section we introduce our notation and give a geometric argument showing that the frame
approach gives the least squares solution for the cm;n in (1.2).
Given g ∈ L2 .R/ and Þ; þ > 0, we shall denote by Tg;Þ;þ the operator mapping f ∈ L2 .R/ to
the sequence f; gmÞ;nþ :
Tg;Þ;þ f = .f; gmÞ;nþ /m;n∈Z :
(2.1)
We shall typically consider Tg;Þ;þ as an operator from L2 .R/ to `2 .Z2 /. While every component
of Tg;Þ;þ defines a bounded linear functional on L2 .R/, putting them together does not always lead
to a bounded operator to `2 .Z2 /. A counterexample is given by, for example, Þ = þ = 1 and g
−5=8
; one then easily checks
piecewise
P constant on2 the intervals [k; k + 1/, with g |[k;k+1/ = .1 + |k|/
that m;n |g; gm;n | diverges. Similar counterexamples can be constructed for other choices of
440
I. Daubechies, H. J. Landau, and Z. Landau
Þ; þ. Note, however, that Tg;Þ;þ is always densely defined (since Tg;Þ;þ f ∈ `2 when f ∈ C0∞ .R/,
regardless of the choice of g ∈ `2 ) and closable (since fn → 0 and Tg;Þ;þ fn → c ∈ `2 implies
ck;` = limn→∞ fn ; gkÞ;`þ = 0). We shall generally restrict ourselves to the case where Tg;Þ;þ is
bounded; all exceptions will be clearly indicated. In order to guarantee that Tg;Þ;þ is bounded, for
any choice of Þ; þ, it is sufficient to require some mild decay for g.x/ or ĝ.¾ /.
2.1.
Proposition
Suppose that g satisfies either |g.x/| ≤ C.1+|x|/−1−ž for all x ∈ R or |ĝ.¾ /| ≤ C.1+|¾ |/−1−ž
for all ¾ ∈ R, with ž > 0. Then, for all Þ; þ > 0, Tg;Þ;þ is bounded from L2 .R/ to `2 .Z2 /.
Proof.
1. With the normalization fˆ.¾ / =
checks that
R
e−2³ ix¾ f .x/ dx for the Fourier transform, one easily
f; gmÞ;nþ = e2³ inmÞþ fˆ; ĝ nþ;−mÞ :
It therefore suffices to discuss only the case where g.x/ decays.
2. Now, for any f ∈ L2 .R/ with compact support
Tg;Þ;þ f 2 =
X
| f; gmÞ;nþ |2
m;n
=
þ
X þZ
þ
þ
m;n
∞
−∞
þ2
þ
e2³ imÞx f .x/ g.x − nþ/ dx þþ
þ
þþ2
X X þþZ 1=Þ
k
k
þ
2³ imÞx
=
e
f x−
g x − − nþ dx þ
þ
þ
þ
Þ
Þ
0
m;n
k
þ
þþ2
Z
1 X 1=Þ þþX
k
k
þ
=
f x−
g x − − nþ þ dx
þ
þ
Þ n 0 þ k
Þ
Þ
þ þ
Z
1 X 1=Þ þþ
k þþ
≤
f
x
−
Þ n;k;` 0 þ
Þ þ
þ þ
þ
þ
þf x − ` þ
þ
Þ þ
þ þ
þ
þ
þg x − k − nþ þ
þ
þ
Þ
þ þ
þ
þ
þg x − ` − nþ þ dx
þ
þ
Þ
Z
þ þ
1 X ∞
m þþ þþ m
þ
þ
=
|f .x/| þf x −
þ þg x − − nþ þ |g.x − nþ/| dx
Þ m;n −∞
Þ
Þ
1
≤
Þ
(
XZ
m
∞
−∞
|f .x/|
2
X
n
þ m þþ
þ
|g.x − nþ/| þg x − nþ −
þ dx
Þ
)1=2
×
441
Gabor Time-Frequency Lattices
(
XZ
þ þ m þþ2 X
m þþ
þ
þ
|g.x − nþ/| þg x − nþ −
þf x −
þ
þ dx
Þ
Þ
−∞
n
m
(Z
1
=
Þ
∞
∞
−∞
(Z
∞
−∞
|f .x/|
|f .y/|
2
X
n
2
X
n
X þþ m þþ
|g.x − nþ/|
þg x − nþ −
þ dx
Þ
m
X þþ m þþ
|g.y − nþ/|
þg y − nþ +
þ dy
Þ
m
)1=2
)1=2
×
)1=2
≤ Cf 2 ;
P
where we have used that k |g.z − k /| ≤ C.1 + | |−1 / because of the decay condition
on g.
3. Since the functions with compact support are dense in L2 .R/, the same bound then holds
for all f in L2 .R/.
2
∗
If Tg;Þ;þ is bounded, then so is its adjoint Tg;Þ;þ
from `2 .Z2 / to L2 .R/. The action of this
adjoint on sequences c ∈ `2 .Z2 / is given by
∗
Tg;Þ;þ
c=
X
cm;n gmÞ;nþ ;
(2.2)
m;n
∗
where the truncated sums for |m|; |n| ≤ K converge in norm to the limit Tg;Þ;þ
c as K → ∞.
The gmÞ;nþ constitute a frame if and only if Tg;Þ;þ is bounded and Tg;Þ;þ f ≥ A1=2 f for
all f . This last statement is equivalent to requiring that Ker Tg;Þ;þ = {0} and that Ran Tg;Þ;þ be a
closed subspace of `2 .Z2 /.
For a given f ∈ L2 .R/, finding the possible choices c ∈ `2 .Z2 / such that (2.2) holds amounts
to finding the sequences c for which
∗
Tg;Þ;þ
c=f :
(2.3)
∗
Clearly, adding any element of Ker Tg;Þ;þ
to an arbitrary solution c of (2.3) gives another solution. We
∗
∗
can “mod out” by Ker Tg;Þ;þ by taking the orthogonal projection of c onto Ran Tg;Þ;þ = .Ker Tg;Þ;þ
/⊥ .
2
This means that we have to identify h ∈ L .R/ so that
c − Tg;Þ;þ h ⊥ Ran Tg;Þ;þ ;
or
∗
Tg;Þ;þ
.c − Tg;Þ;þ h/ = 0 :
∗
This immediately gives f = Tg;Þ;þ
Tg;Þ;þ h; the sequence of coefficients c = Tg; Þ;þ h with minimum
2
` -norm in (2.2) is thus given by
442
I. Daubechies, H. J. Landau, and Z. Landau
.Tg; Þ;þ h/m;n = .Tg;∗ Þ;þ Tg; Þ;þ /−1 f; gmÞ;nþ = f; .Tg;∗ Þ;þ Tg; Þ;þ /−1 gmÞ;nþ :
Let us denote the time-frequency shifts in (1.1) by W .mÞ; nþ/, that is,
[W .p; q/f ].x/ = e−2³ ipx f .x − q/ :
(2.4)
One easily checks that Tg;∗ Þ;þ Tg; Þ;þ commutes with all the W .mÞ; nþ/; it follows that its inverse
commutes with them as well, so that
.Tg;∗ Þ;þ Tg; Þ;þ /−1 W .mÞ; nþ/g = W .mÞ; nþ/g̃ ;
(2.5)
g̃ = .Tg;∗ Þ;þ Tg; Þ;þ /−1 g :
(2.6)
with
We shall call this g̃ the frame dual function for the gmÞ;nþ . The following proposition summarizes
our findings (which can also be found in [3], [4]).
2.2.
Proposition
If the gmÞ;nþ constitute a frame, then the sequences c ∈ `2 .Z2 / for which (2.3) holds are exactly
the elements of Tg̃; Þ;þ f + Ker Tg;∗ Þ;þ , with Tg̃; Þ;þ f the minimum `2 -norm solution.
The argument above is essentially the standard argument for a dual frame construction, going back to [4]. What makes Gabor lattices special is that the commutation of W .mÞ; nþ/ with
Tg;∗ Þ;þ Tg; Þ;þ allows us to write the dual frame as consisting of time-frequency translates of a single
function again, as in (1.1). This does not happen for wavelet frames, for example [3]. We shall come
back to the significance of these commutation rules.
3. The Wexler–Raz Identity
The observation on which the analysis by Wexler and Raz in [5] rests is essentially an identity
that links operators Tg; Þ;þ with their counterparts Tf ;1=þ;1=Þ for the “dual” lattice parameters þ1 ; Þ1 .
(We shall come back to how this “duality” is to be understood.) The derivation in [5], by means
of the Poisson summation formula, is justified only for some functions; in [6] Janssen has given a
different, rigorous proof for some consequences of the original WR identity. We wish here to stick
to the identity itself; our first task is to state and prove it.
443
Gabor Time-Frequency Lattices
3.1.
Theorem
Suppose f; g; h ∈ L2 .R/ and Þ; þ > 0 are such that Tg; Þ;þ , Tf ; Þ;þ , Th; 1=þ;1=Þ are bounded.
Then
Tf∗; Þ;þ Tg; Þ;þ h =
1 ∗
T
Tg; 1=þ;1=Þ f :
Þþ h; 1=þ;1=Þ
(3.1)
Remark. Note that we did not include any assumptions on the boundedness of Tg; 1=þ;1=Þ .
This is because the boundedness of Tg; Þ;þ already implies that Tg; 1=þ;1=Þ is bounded.
We shall prove this theorem in several steps. We start by restricting ourselves to nice h; f .
3.2.
Lemma
Assume that Tg; Þ;þ and Tg; 1=þ;1=Þ are both bounded, and let h; f be compactly supported and
bounded. Then (3.1) holds.
Proof.
1. Take any ' ∈ L2 .R/. Mimicking the computation in part 2 of the proof of Proposition 2.1,
we find
Tf∗; Þ;þ Tg; Þ;þ h; ' = Tg; Þ;þ h; Tf ; Þ;þ '
Z
1 X ∞ m m
h x−
g x − − nþ f .x − nþ/ '.x/ dx ;
Þ m;n −∞
Þ
Þ
=
where the integral and series converge absolutely.
2. Similarly
Th;∗ 1=þ;1=Þ Tg; 1=þ;1=Þ f; '
=þ
XZ
k;`
∞
`
f .x − kþ/ g x − kþ −
Þ
−∞
and (3.1) follows immediately.
`
'.x/ dx ;
h x−
Þ
2
Next we introduce an auxiliary operator to rewrite (3.1) in a different form. Define the operator
R on `2 .Z2 / by
.R c/k;` = e2³ i k` ck;−` ;
444
I. Daubechies, H. J. Landau, and Z. Landau
note that R is unitary and symmetric, R2 = Id, R∗ = R . It is easy to check that if Tf ; Þ;þ and
Tg; Þ;þ are bounded, then
Tf ; Þ;þ g = RÞþ Tg;Þ;þ f :
(3.2)
We can then use the results of Lemma 3.2 to prove the following lemma.
3.3.
Lemma
Assume that Tg; Þ;þ and Tg; 1=þ;1=Þ are bounded, and choose ' in L2 .R/ so that |'.x/| ≤
c.1 + |x|/−1−ž for some ž > 0. Then
∗
T';Þ;þ
RÞþ Tg; Þ;þ =
1 ∗
T
R1=Þþ T';1=þ;1=Þ :
Þþ g;1=þ;1=Þ
(3.3)
Proof.
1. If f; h are bounded and compactly supported, then
Tg; Þ;þ h; Tf ; Þ;þ ' =
1
Tg; 1=þ;1=Þ f; Th; 1=þ;1=Þ ' :
Þþ
=
1
'; Tg;1=þ;1=Þ f ;
T
Þþ h;1=þ;1=Þ
(3.4)
where we have used that
.Tf ; Þ;þ g/∗m;n = .Tf ;Þ;þ g/−m;n :
(3.5)
2. By (3.2) we can rewrite this as
Tg; Þ;þ h; RÞ;þ T';Þ;þ f =
1
R1=Þ;þ T';1=þ;1=Þ h; Tg;1=þ;1=Þ f :
Þþ
Since this now holds for all bounded and compactly supported h and f , (3.3)
follows.
2
We now immediately have
3.4.
Lemma
If Tf ; Þ;þ , Tg; Þ;þ , Th; 1=þ;1=Þ , and Tg; 1=þ;1=Þ are bounded, then (3.1) holds.
Gabor Time-Frequency Lattices
445
Proof. Apply both sides of (3.3) to h, and take the inner product with f . Since Tf ; Þ;þ
and Th; 1=þ;1=Þ are bounded, we can use (3.2) and (3.5) to bring this back to the form (3.4). Since
this is now valid for all ' in the dense set of functions decaying at least as fast as .1 + |x|/−1−ž , we
conclude (3.1).
2
This proves the Wexler–Raz identity, except that we still have to get rid of the separate condition
that Tg; 1=þ;1=Þ be bounded. We first prove the following lemma.
3.5.
Lemma
For arbitrary Þ; þ > 0, define J = J .Þ; þ/ := Þþ, that is, J ∈ N and J − 1 < Þþ ≤ J .
For j = 0; : : : ; J − 1, define
aj = j=Þ;
bj = min
j +1
;þ
Þ
'j .x/ = [aj ;bj / .x/ = 1
that is, bj =
j +1
if j ≤ J − 2; bJ −1 = þ ;
Þ
if aj ≤ x < bj ;
0
otherwise:
Then
J −1
X
j =0
T'∗j ;Þ;þ T'j ;Þ;þ =
1
IdL2 .R/
Þ
(3.6)
and
J −1
X
j =0
T'j ;1=þ;1=Þ T'∗j ;1=þ;1=Þ = þ Id`2 .Z2 / :
Proof.
1. To prove (3.6), it suffices to show that, for all f ∈ L2 .R/,
J −1
X
j =0
T'j ;Þ;þ f 2 =
1
f 2 :
Þ
(3.7)
446
I. Daubechies, H. J. Landau, and Z. Landau
Now
T'j ;Þ;þ
þ
þ2
þ
X þþZ nþ+bj
þ
−2³ imÞx
e
f .x/ dx þ
f =
þ
þ
þ
nþ+a
j
m;n
2
Z
1X
|f .x/|2 [aj ;bj / .x − nþ/ dx :
=
Þ n
Since
J −1
X
[aj ;bj / .y/ = [0;þ/ .y/ ;
(3.8)
j =0
we have
J −1
X
T'j ;Þ;þ
j =0
Z
1X
1
|f .x/|2 [nþ;.n+1/þ/ .x/ dx = f 2 :
f =
Þ n
Þ
2
2. P
Similarly, to prove (3.7), it is sufficient to establish that, for all c ∈P`2 .Z2 /; þc2 =
J −1
2
−2³ imx=þ
,
m cm;n e
j =0 T'j ;1=þ;1=Þ c . If we introduce the þ-periodic function cn .x/ =
then we have
T'∗j ;1=þ;1=Þ
þ2
Z þþX
n þþ
þ
c = þ
c .x/[aj ;bj / x −
þ dx
þ n n
Þ þ
2
=
XZ
n
|cn .x/|2 [aj ;bj / x −
dx ;
Þ
n
ð
Ð
where we have used that, for fixed j , the intervals aj + Þn ; bj + Þn , n ∈ Z, are all disjoint.
It follows then from (3.8) that
J −1
X
j =0
T'∗j ;1=þ;1=Þ c2 =
XZ
n
|cn .x/|2 [0;þ/ x −
dx
Þ
n
=
XZ
0
n
=þ
X
n;m
This now implies the following lemma.
þ
þ
þ2
þX
þ
þ
−2³ im.x+n=Þ/=þ þ
cm;n e
þ
þ dx
þ m
þ
|cm;n e−2³ imn=Þþ |2 = þc2 :
2
447
Gabor Time-Frequency Lattices
3.6.
Lemma
Assume that Tg; Þ;þ and Tg;1=þ;1=Þ are bounded. Let J ≥ 1 and 'j , j = 0; : : : ; J − 1, be defined
as in Lemma 3.5. Then
∗
Tg;1=þ;1=Þ
=Þ
J −1
X
j =0
T'∗j ;Þ;þ RÞþ Tg; Þ;þ T'∗j ;1=þ;1=Þ R1=Þþ :
(3.9)
Apply (3.3) (with ' = 'j ) to Þ T'∗j ;1=þ;1=Þ R1=Þþ c, and sum over j . Then (3.9)
immediately follows from (3.7).
2
Proof.
Formula (3.9) now suggests that Tg;1=þ;1=Þ is defined as a bounded operator whenever Tg; Þ;þ
is. This is in fact the case.
3.7.
Lemma
∗
is given by formula (3.9).
If Tg; Þ;þ is bounded, then so is Tg;1=þ;1=Þ . Moreover, Tg;1=þ;1=Þ
Proof.
1. The right-hand side of (3.9) certainly defines a bounded operator. Let us check how this
operator acts on the “elementary” sequences ek;` ∈ `2 .Z2 / defined by .ek;` /m;n = Žk;m Ž`;n .
One immediately has
T'∗j ;1=þ;1=Þ R1=Þþ ek;` = e−2³ i k`=Þþ T'∗j ;1=þ;1=Þ ek;−`
= e−2³ i k`=Þþ W
k −`
;
'j ;
þ Þ
where the W -operators are as defined in (2.4). Applying (3.2) we then have
RÞþ Tg; Þ;þ T'∗j ;1=þ;1=Þ R1=Þþ ek;` = e−2³ i k`=Þþ TW .−k=þ;−`=Þ/ 'j ;Þ;þ g :
2. Next, observe that W .mÞ; nþ/ commutes with W
that
TW .k=þ;`=Þ/'j ;Þ;þ g
Ð
m;n
−
= W
−k −`
;
þ Þ
k `
;
þ Þ
, for any m; n; k; ` ∈ Z. It follows
×
∗
g; W .mÞ; nþ/'j
− ×
k `
= e2³ i k`=Þþ W
g; W .mÞ; nþ/'j
;
þ Þ
= e2³ i k`=Þþ T'j ;Þ;þ g k=þ;`=Þ :
448
I. Daubechies, H. J. Landau, and Z. Landau
Putting it all together, we have now
Þ T'∗j ;Þ;þ RÞþ Tg; Þ;þ T'∗j ;1=þ;1=Þ R1=Þþ ek;` = Þ T'∗j ;Þ;þ T'j ;Þ;þ g k=þ;`=Þ :
3. Summing over j leads to (use (3.6))
Þ
J −1
X
j =0
∗
T'∗j ;Þ;þ RÞþ Tg;Þ;þ T'∗j ;1=þ;1=Þ R1=Þþ ek;` = ḡ k=þ;`=Þ = Tḡ;1=þ;1=Þ
ek;` ;
which proves our claim.
2
Since Tg;1=þ;1=Þ is obviously bounded if and only if Tg;1=þ;1=Þ is, this shows that we can indeed drop the additional condition that Tg;1=þ;1=Þ be bounded in the statement of Lemma 3.4. This
completes the proof of Theorem 3.1.
Now that we have established (3.1) rigorously, with minimal assumptions on the functions
involved, we can exploit it as in the original paper [5]. Suppose that g b is any “dual function” for
the frame gmÞ;nþ in the sense that Tgb ;Þ;þ is bounded, and Tg;∗ Þ;þ Tgb ;Þ;þ = Id, or, for all f in L2 .R/,
f =
X
b
f; gmÞ;nþ
gmÞ;nþ :
m;n
(The dual frame function g̃ constructed above is an example of such a g b , but it is by no means the
only one in general.) Taking adjoints, we obtain Tg∗b ;Þ;þ Tg; Þ;þ = Id; applying this to h, which decays
like .1 + |x|/−1−ž , and using the WR identity (3.1) we conclude that
∗
Th;1=þ;1=Þ
Tg;1=þ;1=Þ g b = Þþ h :
(3.10)
This means that the sequence c = Tg;1=þ;1=Þ g b should satisfy
n
cm;n e−2³ i .m=þ/x h x −
= Þþ h.x/ ;
Þ
m;n
X
for all h in a dense set. It suffices to choose h.x/ = [0;þ] .x/ to see that this is possible only if
cm;n = Þþ Žm;0 Žn;0 . (Note that Þþ ≤ 1, since we are assuming that the gmÞ;nþ constitute a frame.)
This argument has proved part of the following result.
3.8.
Proposition
Assume that the gmÞ;nþ constitute a frame. Then a function g b such that Tgb ;Þ;þ is bounded
satisfies
∗
Tg;Þ;þ
Tgb ;Þ;þ = Id = Tg∗b ;Þ;þ Tg;Þ;þ
(3.11)
449
Gabor Time-Frequency Lattices
if and only if
Tg;1=þ;1=Þ g b = Þþ e0;0 = Tgb ;1=þ;1=Þ g :
(3.12)
Proof.
We already saw that (3.11) implies (3.12). To prove the converse, observe that
(3.12) immediately implies (3.10) for all h with decay like .1 + |x|/−1−ž . By the WR identity, this
implies Tg∗b ;Þ;þ Tg;Þ;þ h = h for these h, which proves (3.11).
2
Note that (3.12) also implies
Ð
b
Tg;1=þ;1=Þ gk=þ;`=Þ
m;n
−
= g ;W
b
k `
;
þ Þ
∗
W
×
m n
;
g
þ Þ
−
×
m−k n−`
= e−2³ i .k=Þþ/.`−m/ g b ; W
;
g
þ
Þ
= Þþ Žk;m Ž`;n = Þþ.ek;` /m;n :
It follows that Tg;1=þ;1=Þ Tg∗b ;1=þ;1=Þ c = Þþ c for the dense set of c in `2 with finitely many nonzero
entries, or
∗
Tg;1=þ;1=Þ Tg∗b ;1=þ;1=Þ = Þþ Id = Tgb ;1=þ;1=Þ Tg;1=þ;1=Þ
:
(3.13)
This way of rewriting (3.12) enhances the symmetry between (3.11) and (3.12). Note also that for
Þþ ≤ 1, the integer J in Lemmas 3.5 to 3.7 is exactly 1, and the corresponding special function '0
is “self-dual”, in the sense that
T'0 ;1=þ;1=Þ T'∗0 ;1=þ;1=Þ = þ Id;
T'∗0 ;Þ;þ T'0 ;Þ;þ =
1
Id :
Þ
In fact, (3.9) is equally valid if '0 is replaced by a general function h for which the hmÞ;nþ constitute
a frame; if hb is dual to h in the sense of (3.11), one has
∗
∗
Tg;1=þ;1=Þ
= Th∗b ;Þ;þ RÞþ Tg; Þ;þ Th;1=þ;1=Þ
R1=Þþ :
(3.14)
4. The Wexler–Raz Dual Function Is Identical To
the Frame Dual Function
Proposition 3.8 characterizes all the functions g b “dual” to the gmÞ;nþ in the sense of (3.11) as
the pre-images of Þþ e0;0 under the map Tg;1=þ;1=Þ . Among all these dual functions g b we can pick the
one with minimal norm; we shall call this the Wexler–Raz dual function (as opposed to the frame dual
450
I. Daubechies, H. J. Landau, and Z. Landau
function g̃ at the end of §2 ) and denote it by g # . We can find an explicit expression for g # by an
argument similar to what led to (2.6) for g̃. Again, the minimal norm solution g b to Tg;1=þ;1=Þ g b =
Þþ e0;0 will result if we “mod out” by Ker Tg;1=þ;1=Þ by taking the orthogonal projection of an arbitrary
∗
∗
solution g b onto .Ker Tg;1=þ;1=Þ /⊥ = Ran Tg;1=þ;1=Þ
. We have used here that the range of Tg;1=þ;1=Þ
is a closed set, which follows from (3.13), since, for any dual function g b ,
c =
1
1
∗
∗
Tgb ;1=þ;1=Þ Tg;1=þ;1=Þ
Tgb ;1=þ;1=Þ Tg;1=þ;1=Þ
c ≤
c :
Þþ
Þþ
(4.1)
∗
To find the projection of g b onto Ran Tg;1=þ;1=Þ
, we have to identify c in `2 .Z2 / so that g b −
∗
∗
∗
b
Tg;1=þ;1=Þ c ⊥ Ran Tg;1=þ;1=Þ or Tg;1=þ;1=Þ .g − Tg;1=þ;1=Þ
c/ = 0. This is immediately equivalent
∗
∗
has a bounded inverse by (4.1), c =
with Tg;1=þ;1=Þ Tg;1=þ;1=Þ c = Þþ e0;0 or, since Tg;1=þ;1=Þ Tg;1=þ;1=Þ
∗
∗
−1
b
c=
Þþ.Tg;1=þ;1=Þ Tg;1=þ;1=Þ / e0;0 . The projection of g onto .Ker Tg;1=þ;1=Þ /⊥ is therefore Tg;1=þ;1=Þ
∗
∗
−1
Þþ Tg;1=þ;1=Þ .Tg;1=þ;1=Þ Tg;1=þ;1=Þ / e0;0 . The following proposition summarizes these findings.
4.1.
Proposition
Assume that the gmÞ;nþ constitute a frame. Then the functions g b for which Tg;1=þ;1=Þ g b =
Þþ e0;0 are exactly the elements of g # + Ker Tg;1=þ;1=Þ , with
∗
∗
g # = Þþ Tg;1=þ;1=Þ
.Tg;1=þ;1=Þ Tg;1=þ;1=Þ
/−1 e0;0 ;
(4.2)
g # itself is the solution with minimum L2 -norm.
Even though g̃ and g # are obtained as the solutions of different minimization problems, their
geometric interpretation as the result of projecting arbitrary c solving (2.3) or arbitrary g b solving
(3.12) now easily leads to the following result.
4.2.
Proposition
Assume that the gmÞ;nþ constitute a frame. Then the frame dual function g̃ defined by (2.6) and
the Wexler–Raz dual function g # defined by (4.2) are identical.
Proof.
1. We start by rederiving in a few lines that the g̃ mÞ;nþ are a frame with frame dual function
g. By the definition of g̃, we have
g̃ = .Tg;∗ Þ;þ Tg; Þ;þ /−1 g
or
g = Tg;∗ Þ;þ Tg; Þ;þ g̃ :
(4.3)
Gabor Time-Frequency Lattices
451
It follows easily that
W .mÞ; nþ/g = .Tg;∗ Þ;þ Tg; Þ;þ / W .mÞ; nþ/ g̃ ;
hence
∗
Tg;∗ Þ;þ = Tg;∗ Þ;þ Tg; Þ;þ Tg̃;Þ;þ
:
(4.4)
Consequently,
∗
Tg;∗ Þ;þ Tg; Þ;þ = Tg;∗ Þ;þ Tg; Þ;þ Tg̃;Þ;þ
Tg̃;Þ;þ Tg;∗ Þ;þ Tg; Þ;þ
or
∗
.Tg;∗ Þ;þ Tg; Þ;þ /−1 = Tg̃;Þ;þ
Tg̃;Þ;þ :
(4.5)
∗
g̃ = Tg̃;Þ;þ
Tg̃;Þ;þ g :
(4.6)
This implies, by (4.3),
2. Since Tg̃;Þ;þ is bounded by (4.5), we can apply the WR identity to (4.6) and find
g̃ =
1 ∗
Tg̃;1=þ;1=Þ g̃ :
T
Þþ g;1=þ;1=Þ
(4.7)
∗
∗
This implies that g̃ is an element of Ran Tg;1=þ;1=Þ
. Its projection onto Ran Tg;1=þ;1=Þ
is
therefore again g̃. On the other hand, g̃ is a dual function for the gmÞ;nþ in the sense of
∗
(3.11), implying that it solves (3.12). By Proposition 4.1, its projection onto Ran Tg;1=þ;1=Þ
is therefore equal to g # . It follows that g # = g̃, as claimed.
2
Our analysis so far has shown that if the gmÞ;nþ constitute a frame, then
1:
Tg;1=þ;1=Þ
2:
∗
Tg;1=þ;1=Þ Tg;1=þ;1=Þ
3:
∗
∗
g̃ = Þþ Tg;1=þ;1=Þ
.Tg;1=þ;1=Þ Tg;1=þ;1=Þ
/−1 e0;0 :
is bounded:
is invertible and has a bounded inverse:
(4.8)
(4.9)
(4.10)
Note that this WR construction of the dual function g # works whenever (4.8)–(4.10) hold; no a
priori assumption that the gmÞ;nþ constitute a frame needs to be made. In order to establish complete
452
I. Daubechies, H. J. Landau, and Z. Landau
equivalence between the frame dual function and the WR dual function, it remains to show that
(4.8)–(4.10) also imply that the gmÞ;nþ constitute a frame. This essentially follows from the results
proved in §3. We already know that Tg;Þ;þ is bounded if and only if Tg;1=þ;1=Þ is (by Lemma 3.7).
∗
We therefore only need to prove that if Tg;1=þ;1=Þ Tg;1=þ;1=Þ
has a bounded inverse (necessary to make
∗
(4.10) work), then Tg;Þ;þ Tg;Þ;þ likewise has a bounded inverse. By interchanging the roles of .Þ; þ/
and . þ1 ; Þ1 /, in Lemma 3.5, we construct K ≥ 1 and functions k , k = 0; : : : ; K − 1, such that
K−1
X
T ∗k ;1=þ;1=Þ T
k ;1=þ;1=Þ
= þ IdL2 .R/ ;
k=0
K−1
X
T
k ;Þ;þ
T ∗k ;Þ;þ =
k=0
1
Id 2 2 :
Þ ` .Z /
(4.11)
By (3.3) we have
T ∗k ;Þ;þ RÞþ Tg;Þ;þ =
1 ∗
R1=Þþ T
T
Þþ ḡ;1=þ;1=Þ
k ;1=þ;1=Þ
:
Multiplying each side on the left with its adjoint and summing over k leads to
∗
Tg;Þ;þ
Tg;Þ;þ =
X
1 K−1
∗
T ∗k ;1=þ;1=Þ R1=Þþ Tḡ;1=þ;1=Þ Tḡ;1=þ;1=Þ
R1=Þþ T
2
Þþ k=0
k ;1=þ;1=Þ
:
For all f ∈ L2 .R/ we therefore have
Tg;Þ;þ f 2 =
X
1 K−1
T ∗
R1=Þþ T
2
Þþ k=0 ḡ;1=þ;1=Þ
k ;1=þ;1=Þ
2
f :
(4.12)
If now, for all c ∈ `2 .Z2 /,
∗
Ac2 ≤ Tḡ;1=þ;1=Þ
c2 ≤ Bc2 ;
∗
with A > 0, B < ∞ (that is, if Tg;1=þ;1=Þ is bounded and Tg;1=þ;1=Þ Tg;1=þ;1=Þ
has a bounded inverse),
then (4.12) implies
A
B
f 2 ≤ Tg;Þþ f 2 ≤
f 2 ;
Þþ
Þþ
P
where we have used that K−1
k=0 T
summarizes all our findings.
k ;1=þ;1=Þ
f 2 = þf 2 , by (4.11). The following theorem
Gabor Time-Frequency Lattices
4.3.
453
Theorem
For g ∈ L2 .R/, Þ; þ > 0, the operator Tg; Þ;þ defined by (2.1) is bounded from L2 .R/ to
` .Z / if and only if Tg;1=þ;1=Þ is. Moreover, Tg;∗ Þ;þ Tg; Þ;þ has a bounded inverse if and only if
∗
Tg;1=þ;1=Þ Tg;1=þ;1=Þ
has a bounded inverse, and then the “frame dual function” g̃ and the “Wexler–
Raz dual function” g # coincide, that is,
2
2
∗
∗
.Tg;∗ Þ;þ Tg; Þ;þ /−1 g = Þþ Tg;1=þ;1=Þ
.Tg;1=þ;1=Þ Tg;1=þ;1=Þ
/−1 e0;0 ;
where e0;0 ∈ `2 .Z2 / is the sequence .e0;0 /k;` = Žk;0 Ž`;0 .
The main practical interest of this result is that it leads to a simpler construction of the dual
2
function g̃ than the recursion proposed in [3]. For the case g.x/ = ³ −1=4 e−x =2 , Þ = 0:25, þ = 2:0,
for instance, one finds A 1:600, B 2:425 (see [3]; more accurate values can be found in [9]),
and the frame dual function g̃ is then computed as
k
∞ X
2
2
∗
Id −
Tg;1=4;2 g = lim g K ;
T
g̃ =
K→∞
A + B k=0
A + B g;1=4;2
where g K =
2
A+B
PK
k=0 .Id
g K+1 =
−
2
T∗
A+B g;1=4;2
Tg;1=4;2 /k g can also be defined recursively by
X
2
2
g + gK −
g K ; gm=4;2n gm=4;2n
A+B
A + B m;n
and
g0 =
2
g:
A+B
If one writes
gK =
X
½K
m;n gm=4;2n ;
m;n
then this leads to a recursive computation of the ½K
m;n , which converges exponentially fast in K,
ÐK
K
g̃ − g K ≤ C B−A
C.0:2/
.
The
computation
of g # can be carried out by exactly the same
B+A
trick, with
∗
.Tg;1=2;4 Tg;1=2;4
/−1 e0;0
k
∞ X
4
4
∗
Id −
=
e0;0 = lim eK ;
Tg;1=2;4 Tg;1=2;4
K→∞
A + B k=0
A+B
454
I. Daubechies, H. J. Landau, and Z. Landau
4
4
4
∗
where e0 = A+B
e0;0 and eK = A+B
e0;0 + eK − A+B
Tg;1=2;4 Tg;1=2;4
eK . This amounts to writing g #
P
K
K
as a limit of m;n ¼m;n gm=2;4n , where the ¼m;n obey a recursion similar to that of the ½K
m;n above. For
1
the same precision, we need however to compute fewer ¼K
(about
as
many)
than
½K
m;n
m;n , and the
4
recursion will also use smaller matrices because gm=2;4n ; g decays faster in m; n than gm=4;2n ; g.
Remark.
The dual function g̃ = g # can be used to expand arbitrary functions into the
gmÞ;nþ , as in
f =
X
f; g̃ mÞ;nþ gmÞ;nþ :
m;n
In the case Þþ = 12 , as in the example above, it is also possible to regroup the gmÞ;nþ and g−mÞ;nþ into
new functions E|m|Þ;nþ , that are then independent. The expansion of f into this basis E|m|Þ;nþ uses
coefficients that can again be computed by means of inner products of f with the g̃ mÞ;nþ . This gives
another use for dual frame functions, allowing, for example, expansions in Gaussian basis functions,
without redundancy. For details, see [10].
5. An Independent Proof
Here we return to §4, proving the identity of the two minimizing dual functions, g̃ and g # ,
directly from basic considerations, without recourse to the Wexler–Raz formula. Accordingly, we
assume that gmÞ;nþ constitute a frame and, as in (2.6), define
∗
g̃ = .Tg;Þ;þ
Tg;Þ;þ /−1 g :
The extremal property of g̃ is that (1.6) generates the expansion coefficients in (1.2) of least `2 norm.
On combining (1.6) and (1.2) we see that g̃ satisfies
∗
Tg;Þ;þ
Tg̃;Þ;þ = I :
However, the equation
∗
Tg;Þ;þ
Tgb ;Þ;þ = I
can also have other solutions g b , and we denote by g # that solution having least L2 norm.
Any two g b in (5.1) differ by u for which
∗
Tg;Þ;þ
Tu;Þ;þ = 0
(5.1)
Gabor Time-Frequency Lattices
455
or, equivalently, one for which
Tu;Þ;þ p; Tg;Þ;þ q = 0 ;
for all p; q ∈ L2 . Since, as we have seen, there is a unitary map between Tu;Þ;þ p and Tp̄;Þ;þ ū, it
follows that
Tp;Þ;þ u; Tq;Þ;þ g = 0
or, finally,
∗
u; Tp;Þ;þ
Tq;Þ;þ g = 0 :
We conclude that g # is the (unique) projection of g b onto the subspace to which u is orthogonal, that
∗
Tq;Þ;þ g. But
is, the unique solution of (5.1) that lies in the subspace G generated by functions Tp;Þ;þ
as in (4.5),
∗
∗
.Tg;Þ;þ
Tg;Þ;þ /−1 = Tg̃;Þ;þ
Tg̃;Þ;þ ;
whence
∗
g̃ = Tg̃;Þ;þ
Tg̃;Þ;þ g ∈ G :
Consequently g # = g̃, as was to be proved.
6. von Neumann Algebras
The operator families W .j Þ; kþ/ and W . mþ ; Þn / linked by the Wexler–Raz formula are distinguished by commuting with one another. This and some of our earlier proofs point to commutativity
as an important factor here and thereby suggest a possible connection with von Neumann algebras,
which have commutativity at their core. Indeed, these algebras have already been successfully applied to time-frequency lattices, for until the recent proof [11] they gave the only means to show
that for the span of W .j Þ; kþ/g to be dense in L2 .R/, it was necessary that Þþ ≤ 1. However, this
argument had relied on the coupling constant, a feature of these algebras that is hard to establish.
We will sketch the basic facts of the subject—an excellent reference is [12]—and show that
it offers a clear view of the Wexler–Raz identity and of its properties. Further, we will establish
the above density condition in a simple and direct way. Finally, we will also use the Wexler–Raz
formula to give an elementary proof of the existence of the coupling constant for von Neumann
algebras associated with time-frequency lattices.
456
I. Daubechies, H. J. Landau, and Z. Landau
Definition. Let B.H / denote the space of bounded linear operators on a Hilbert space H .
Suppose that a given set of operators T ⊂ B.H / includes the identity as well as the adjoint T ∗ of
every T ∈ T . Let the commutant algebra A ⊂ B.H / be the collection of bounded operators that
commute with every T ∈ T , and denote by {{T }} the linear combinations of a finite number of finite
products of Ti ∈ T . The aim is to add to {{T }} all the bounded operators that commute with A .
By the double commutant theorem [12], this can be done by forming the closure A of {{T }} with
respect to either of:
• strong convergence: A ∈ A if there is a sequence (or net) Ti ∈ {{T }} with Ti .h/ → A.h/
for each h ∈ H ;
• weak convergence: A ∈ A if there is a sequence (or net) Ti ∈ {{T }} with .Ti h1 ; h2 / →
.Ah1 ; h2 / for each h1 ; h2 ∈ H .
A is termed a von Neumann algebra. Clearly, A commutes also with A.
Here, let T consist of the frequency and time translations W .j Þ; kþ/, −∞ < j; k < ∞, acting
on L2 .R/, as defined in (2.4),
W .j Þ; kþ/f = e−2³ iÞj t f .t − kþ/ :
(6.1)
W .Þ; 0/W .0; þ/ = e2³ iÞþ W .0; þ/W .Þ; 0/
(6.2)
The commutation relation
reduces all finite products of the generators to the form e2³ iÞþM W .j Þ; kþ/ for suitable integers
M; j; k. Since W .j Þ; kþ/ is unitary,
W .j Þ; kþ/∗ = [W .j Þ; kþ/]−1 = e−2³ iÞþj k W .−j Þ; −kþ/ ;
(6.3)
so T contains the adjoints of its elements as well as I = W .0; 0/, as required.
The elements W . mþ ; Þn / clearly commute with A and so belong to A . A
straightforward argument (Appendix 6.1) shows that A is generated by {{W . mþ ; Þn /}}.
Commutant.
Trace.
A faithful trace is a linear functional on B.H / with the properties
tr .I / = 1 ;
tr .AB/ = tr .BA/ ;
tr .A∗ A/ > 0 except when A = 0.
(6.4)
Gabor Time-Frequency Lattices
457
In the algebra {{W .j Þ; kþ/}}, the elements consist of finite linear combinations
T =
X
cj k W .j Þ; kþ/ :
|j;k|<N
Using (6.3), we can verify by explicit calculation that the choice
tr .T / ≡ c00
(6.5)
has the required properties.
Let i , i = 1; : : : ; K, denote the characteristic functions of disjoint contiguous intervals, none
longer than min. Þ1 ; þ/, that together make up the interval [0; Þ1 ]. Then we have
trA .T / = Þ
K
X
.T i ; i / :
(6.6)
i=1
For it is sufficient to check (6.6) for T = W .j Þ; kþ/. Since the intervals are no longer than þ, any
two translates of one of them by multiples of þ are disjoint. Thus the only nonzero contribution to
(6.6) comes from k = 0, whereupon the sum in (6.6) becomes
Z
Þ
1=Þ
e2³ iÞj t dt = 0;
j = 0 :
0
The right-hand side of (6.6) allows the trace to be extended to A by continuity.
When Þþ is irrational, the trace is unique. For if j = 0, the requirement tr AB = tr BA,
together with the commutation relation (6.2), yield
tr W .j Þ; kþ/ = tr W .j Þ; .k − 1/þ/W .0; þ/
= tr W .0; þ/W .j Þ; .k − 1/þ/ = tr e−2³ iÞþj W .j Þ; kþ/ ;
implying that trW .j Þ; kþ/ = 0; while if j = 0, the same argument applies to
tr W .0; kþ/ = tr W .−Þ; 0/W .0; kþ/W .Þ; 0/ = tr e−2³ iÞþk W .0; kþ/ :
A New Hilbert Space. We can now use the trace to define a scalar product on the algebra
{{W .j Þ; kþ/}}, viewed as a linear space, by
[A; B] ≡ trA B ∗ A :
(6.7)
Since each A ∈ A is a weak limit of operators from the algebra, the expression (6.6) shows that it
is likewise a limit in the norm defined by (6.7). Thus the completion of {{W .j Þ; kþ/}} in the scalar
458
I. Daubechies, H. J. Landau, and Z. Landau
product (6.7) yields a Hilbert space (of operators) that includes A and which we therefore denote by
L2 .A/. There is an isomorphism between A acting on the original H, and its acting on L2 .A/ by left
multiplication; this identification is known as the G–N–S (Gelfand–Naimark–Segal) construction.
An important fact for us is that, by (6.5), the elements W .j Þ; kþ/ constitute an orthonormal basis
for L2 .A/. Of course, the identical considerations apply to A and give W . mþ ; Þn / as an orthonormal
basis for L2 .A /.
As heretofore, we assume f; g ∈ L2 , and suppose initially that Tf ;Þ;þ and Tg;Þ;þ are bounded
operators. We have already seen that Tf∗;Þ;þ Tg;Þ;þ commutes with W .j Þ; kþ/, hence
Tf∗;Þ;þ Tg;Þ;þ ∈ A :
To find its trace in A we use (6.6) with Þ replaced by
trA .Tf∗;Þ;þ Tg;Þ;þ / =
1
þ
to reflect the change of algebra, obtaining
1X ∗
1X
Tf ;Þ;þ Tg;Þ;þ j ; j =
Tg;Þ;þ j ; Tf ;Þ;þ j ;
þ j
þ j
(6.8)
with j characteristic functions of intervals smaller than Þ1 that decompose [0; þ]. For each k, the
components cj;k of Tf ;Þ;þ j are the Fourier coefficients in the basis {e−2³ iÞj t } of f .t/j .t − kþ/, a
function supported on an interval no longer than Þ1 . By Parseval’s theorem,
Tg;Þ;þ j ; Tf ;Þ;þ j =
1X
g.t/ j .t + kþ/; f .t/;
Þ k
and on summing these components over j we find
trA .Tf∗;Þ;þ Tg;Þ;þ / =
1 X
1
g [0;þ] .t + kþ/; f =
f; g :
Þþ k
Þþ
(6.9)
Expanding Tf∗;Þ;þ Tg;Þ;þ ∈ A ⊂ L2 .A / in the above basis, we obtain
Tf∗;Þ;þ Tg;Þ;þ
=
X
Tf∗;Þ;þ Tg;Þ;þ ; W
j k
;
þ Þ
½
W
A
j k
;
þ Þ
;
(6.10)
the sum converging in the norm of L2 .A /. Denote the coefficients in (6.10) by j;k ; to evaluate
them, we have by (6.7) and (6.4),
½
j k
j k
j ;k ≡ Tf∗;Þ;þ Tg;Þ;þ ; W
= trA Tf∗;Þ;þ Tg;Þ;þ W ∗
;
;
:
þ Þ A
þ Þ
For an operator S ∈ A , and any u ∈ L2 , by definition
ý
ý
T';Þ;þ Su = {Su; W .mÞ; nþ/'} = u; S ∗ W .mÞ; nþ/' = u; W .mÞ; nþ/S ∗ ' :
(6.11)
Gabor Time-Frequency Lattices
459
This means that
T';Þ;þ S ≡ TS ∗ ';Þ;þ
(6.12)
are merely different ways of writing the same operator. Applying this to (6.11) with S = W ∗ . þj ; Þk /
and using (6.9) we find
−
×
j k
j;k = trA Tf∗;Þ;þ TW .j=þ;k=Þ/g;Þ;þ = .Þþ/−1 f; W
;
g :
þ Þ
Consequently,
{j;k } = .Þþ/−1 Tg;1=þ;1=Þ f ;
(6.13)
that is, f is in the domain of the operator .Þþ/−1 Tg;1=þ;1=Þ and is taken by it to {j;k }; being the
coefficients of an orthonormal expansion, this sequence is square-summable. We note that this
holds as soon as the product Tf∗;Þ;þ Tg;Þ;þ is bounded, without requiring boundedness of the factors
separately.
We can use the unitary map between the Hilbert spaces L2 .A /
and ` , given by the coefficients of an element in L2 .A / in the basis {W . mþ ; Þn /}, to carry an operator
on L2 .A / into its correspondent on `2 . In Appendix 6.2 we show that this map leads to a conjugate∗
linear algebra isomorphism carrying Tf∗;Þ;þ Tg;Þ;þ acting on L2 .A / to .Þþ/−1 Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
2
acting on ` . When composed with the G–N–S construction, this produces a conjugate-linear iso∗
acting on `2 . As any algebra
morphism taking Tf∗;Þ;þ Tg;Þ;þ acting on L2 to .Þþ/−1 Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
isomorphism preserves the spectrum, and as the spectrum of an element is independent of the C ∗
algebra that contains it [1, p. 241], it follows that the spectrum of one of these operators is the
complex conjugate of that of the other. When f = g, this shows that
A Unitary Equivalence.
2
ý
ý
∗
∗
spectrum .Þþ/Tg;Þ;þ
Tg;Þ;þ = spectrum Tg;1=þ;1=Þ Tg;1=þ;1=Þ
;
implying that
∗
∗
∗
.Þþ/Tg;Þ;þ
Tg;Þ;þ = Tg;1=þ;1=Þ Tg;1=þ;1=Þ
= Tg;1=þ;1=Þ
Tg;1=þ;1=Þ ;
(6.14)
whence
p
Þþ Tg;Þ;þ = Tg;1=þ;1=Þ :
This recovers Theorem 4.3.
The Wexler–Raz Identity. Formally, the Wexler–Raz formula comes from applying to
h ∈ L2 the orthonormal expansion (6.10) and (6.13), rewritten here for convenience
460
I. Daubechies, H. J. Landau, and Z. Landau
Tf∗;Þ;þ Tg;Þ;þ =
X
j;k W
j k
;
þ Þ
;
{j;k } = .Þþ/−1 Tg;1=þ;1=Þ f:
∗
For the left-hand side of (6.10) is then Tf∗;Þ;þ Tg;Þ;þ h, while the right-hand side is Th;1=þ;1=Þ
{j k } =
−1 ∗
.Þþ/ Th;1=þ;1=Þ Tg;1=þ;1=Þ f . The equality of these quantities, suggested by (6.10), gives the Wexler–
Raz formula. However, the technical point to be established is that (6.10), which refers to convergence
in trace norm, implies the corresponding equality
∗
Tf∗;Þ;þ Tg;Þ;þ h = .Þþ/−1 Th;1=þ;1=Þ
Tg;1=þ;1=Þ f
(6.15)
in L2 . We prove this in Appendix 6.3.
Alternatively, let V denote the unitary map sending Tf∗;Þ;þ Tg;Þ;þ ∈ L2 .A / onto its coefficients
{j;k }. Suppose to start that u and ' have compact support, and consider
∗
V Tf∗;Þ;þ Tg;Þ;þ ; V T';Þ;þ
Tu;Þ;þ =
=
1
Tg;1=þ;1=Þ f; Tu;1=þ;1=Þ '
.Þþ/2
1
T ∗
Tg;1=þ;1=Þ f; ' ;
.Þþ/2 u;1=þ;1=Þ
(6.16)
the first equality by (6.13). At the same time, since V is unitary,
∗
∗
V Tf∗;Þ;þ Tg;Þ;þ ; V T';Þ;þ
Tu;Þ;þ = [Tf∗;Þ;þ Tg;Þ;þ ; T';Þ;þ
Tu;Þ;þ ]L2 .A /
∗
≡ trA Tu;Þ;þ
T';Þ;þ Tf∗;Þ;þ Tg;Þ;þ :
(6.17)
By (6.12) with S = Tf∗;Þ;þ Tg;Þ;þ and (6.9),
∗
T';Þ;þ Tf∗;Þ;þ Tg;Þ;þ =
trA Tu;Þ;þ
1
1
∗
Tf ;Þ;þ ' =
u; Tg;Þ;þ
T ∗ Tg;Þ;þ u; ' ; (6.18)
Þþ
Þþ f ;Þ;þ
and by comparing (6.17) and (6.18) with (6.16) we find that, as elements of L2 ,
∗
Tf∗;Þ;þ Tg;Þ;þ u = .Þþ/−1 Tu;1=þ;1=Þ
Tg;1=þ;1=Þ f :
(6.19)
The argument of Appendix 6.3, beginning at (A3.3), extends this equality to all u ∈ L2 , thereby
again proving the Wexler–Raz identity (6.15).
In (6.15) we have a slight generalization of Theorem 3.1, for Th;1=þ;1=Þ is not required to be
bounded; the formula asserts that if Tf∗;Þ;þ Tg;Þ;þ is bounded, the sequence Tg;1=þ;1=Þ f is in the domain
∗
and is taken by it into .Þþ/Tf∗;Þ;þ Tg;Þ;þ h.
of Th;1=þ;1=Þ
Gabor Time-Frequency Lattices
461
The preceding arguments apply without modification when only Tf∗;Þ;þ Tg;Þ;þ
is bounded, f; g ∈ L , without requiring boundedness for Tf ;Þ;þ and Tg;Þ;þ individually. Here we
mean that for some K and each compactly supported u ∈ L2 , the sequence Tg;Þ;þ u, known to be
square-summable, lies in the domain of Tf∗;Þ;þ and that Tf∗;Þ;þ .Tg;Þ;þ u/ ≤ Ku. The operator
Tf∗;Þ;þ Tg;Þ;þ is then defined and bounded on a dense subset of L2 and can be extended to all of L2
by continuity. Then for v ∈ L2 of compact support,
An Extension.
2
Tf∗;Þ;þ Tg;Þ;þ u; v = Tg;Þ;þ u; Tf ;Þ;þ v ;
(6.20)
both sides being well defined, so that the trace formula (6.9) continues to hold. Moreover, since from
(6.20),
|Tg;Þ;þ u; Tf ;Þ;þ v| ≤ .Kv/ u ;
the scalar product on the left is a bounded linear functional of u; hence Tf ;Þ;þ v lies in the domain of
∗
Tg;Þ;þ
, and
∗
Tg;Þ;þ
Tf ;Þ;þ v ≤ Kv :
∗
Consequently Tg;Þ;þ
Tf ;Þ;þ is likewise bounded, and again from (6.20)
∗
.Tf∗;Þ;þ Tg;Þ;þ /∗ = Tg;Þ;þ
Tf ;Þ;þ ;
so that (6.19) follows as before. We conclude that the Wexler–Raz identity (6.15) remains valid
whenever the operator on either side is bounded.
Lattice Density. We now give an elementary argument for the heretofore difficult result
than if gmÞ;nþ span L2 the lattice .mÞ; nþ/ must be sufficiently dense.
6.1.
Theorem
If for some g ∈ L2 the functions W .j Þ; kþ/g span L2 , then Þþ ≤ 1.
We want to consider arbitrary g ∈ L2 so that Tg;Þ;þ may be unbounded. We have
seen in §2, however, that this operator is defined and closable in C0∞ .R/, and we denote its closure
∗
Tg;Þ;þ is selfadjoint and bounded below away
by Tg;Þ;þ as well. With ž > 0, the operator žI + Tg;Þ;þ
from zero and, hence, invertible. Let
Proof.
∗
pž = .žI + Tg;Þ;þ
Tg;Þ;þ /−1 g :
∗
∗
Clearly, Tg;Þ;þ
Tg;Þ;þ .žI + Tg;Þ;þ
Tg;Þ;þ /−1 is bounded and, by [13, p. 307], so is Tg;Þ;þ .žI +
∗
∗
−1
Tg;Þ;þ and of Tg;Þ;þ . We have
Tg;Þ;þ Tg;Þ;þ / . Consequently, pž is in the domain of Tg;Þ;þ
462
I. Daubechies, H. J. Landau, and Z. Landau
∗
Tg;Þ;þ pž ;
g = žpž + Tg;Þ;þ
and on forming the scalar product with pž we obtain
žpž 2 + Tg;Þ;þ pž 2 = g; pž :
(6.21)
Setting ¹ = g; pž , it follows from (6.21) that
¹ ≥ Tg;Þ;þ pž 2 ;
∗
while ¹ = pž ; Tg;Þ;þ
e00 = Tg;Þ;þ pž ; e00 , so that
Tg;Þ;þ pž 2 ≥ ¹ 2 ;
it follows that ¹ ≤ 1. (By way of intuition, Tg;Þ;þ pž is constructed to approximate the projection of
∗
∗
e00 onto the range of Tg;Þ;þ .) Now for the bounded operator Tg;Þ;þ
Tg;Þ;þ .žI + Tg;Þ;þ
Tg;Þ;þ /−1 we
find from (6.12) and the definition of pž that
∗
∗
∗
Tg;Þ;þ
Tg;Þ;þ .žI + Tg;Þ;þ
Tg;Þ;þ /−1 = Tg;Þ;þ
Tpž ;Þ;þ ;
whence by using (6.9), extended as in the preceding section, to express the trace of the right-hand
side,
∗
∗
Þþ trA Tg;Þ;þ
Tg;Þ;þ .žI + Tg;Þ;þ
Tg;Þ;þ /−1 = g; pž = ¹ ≤ 1 :
(6.22)
By the functional calculus for selfadjoint operators [13, p. 341], according to which an operator is
represented as multiplication by x on its spectrum, as ž → 0 the operator on the left-hand side of
∗
(6.22) approaches the projection onto the range of Tg;Þ;þ
Tg;Þ;þ , which is dense in L2 whenever that
∗
of Tg;Þ;þ is. Thus the limit of the trace in (6.22) is tr I = 1, whence Þþ ≤ 1.
This completes the proof of Theorem 6.1.
2
The Coupling Constant. The argument in Theorem 6.1 makes no use of the coupling
constant, on which an earlier proof had depended. The existence of this constant is a feature of
certain von Neumann algebras that is difficult to establish. Here we proceed in the other direction
and use some of our earlier considerations to give a simple proof of the existence of the coupling
constant for the algebra A.
6.2.
Theorem
With g ∈ L2 .R/, let Y denote the closure of the subspace of L2 .R/ generated by the elements
of Ag, and let P be the orthogonal projection of L2 .R/ onto Y. Analogously, let Y be the closure
of A g and P be the projection onto Y . Then P ∈ A , P ∈ A, and
463
Gabor Time-Frequency Lattices
trA .P /
1
=
;
Þþ
trA .P /
independently of the choice of g. The above quotient is called the coupling constant for A .
We first verify that P ∈ A , as claimed. Let f = u + v be a decomposition of
f ∈ L into components in Y and in S, its orthogonal complement. Since each W .mÞ; nþ/ maps
Ag onto itself, it does the same with Y and S. It follows that Wf = W u + W v is the corresponding
decomposition of Wf ; hence P Wf = W u = W Pf . Thus P commutes with A; hence P ∈ A .
Analogously, P ∈ A.
∗
∗
and of Tg;Þ;þ
Tg;Þ;þ have the
Suppose first that Tg;Þ;þ is bounded. As the ranges of Tg;Þ;þ
same orthogonal complement, their closures coincide and consist of Y. To simplify notation, set
∗
∗
S = Tg;Þ;þ
Tg;Þ;þ and Q = Tg;1=þ;1=Þ
Tg;1=þ;1=Þ ; we recall that S ∈ A while Q ∈ A. Again by the
functional calculus, the projection P onto Y = range S is given by
Proof.
2
(
S
I− I−
S
P = lim
k→∞
k )
since 1 − .1 − x/k for x ∈ [0; 1] has limit 0 at x = 0 and 1 otherwise, while x = 0 and x = 0
correspond in the spectral decomposition to the null space and range of S, respectively. Thus
trA P = lim
k→∞
k
X
j =1
.−1/j +1
k trA S j
:
j Sj
(6.23)
The analogous formula applies to trA P . Now our preceding results allow us to compare traces and
norms of S and Q. For by the Wexler–Raz identity
Sg =
1
Qg ;
Þþ
whence for k > 1, because S and Q commute,
S k g = S k−1 Sg =
1 k−1
1
Qg =
S
QS k−1 g ;
Þþ
Þþ
so by induction
Sk g =
Thereupon by (6.12) and (6.9)
1
Qk g :
.Þþ/k
(6.24)
464
I. Daubechies, H. J. Landau, and Z. Landau
∗
∗
trA S k = trA S.S k−1 / = trA Tg;Þ;þ
Tg;Þ;þ .S k−1 / = trA Tg;Þ;þ
TS k−1 g;Þ;þ
=
1
1
g; Qk−1 g ;
g; S k−1 g =
Þþ
.Þþ/k
(6.25)
the last equality by (6.24). By the analogous argument applied to Q (in which Þ; þ are replaced by
1=þ; 1=Þ), stopping at the next-to-last of the above equalities yields
trA Qk = Þþ.g; Qk−1 g/ :
(6.26)
Finally by (6.14)
S =
1
Q ;
Þþ
hence from (6.25) and (6.26)
trA S j =Sj =
1
trA Qj =Qj
Þþ
for each j . Consequently
k Ð
j
j
j =1 j trA S =S
Pk k Ð
j
j
j =1 j trA Q =Q
Pk
=
1
;
Þþ
which is then also the value of the limit as k → ∞, which equals the quotient of traces.
To complete the argument, we show next that for any g ∈ L2 the traces trA P and trA P can
be approximated arbitrarily well by corresponding traces of projections Ph and Ph onto subspaces
Ah and A h, with some h for which Th;Þ;þ is bounded. For by (6.6), to approximate the traces it
is sufficient to approximate P and P for a finite number of suitable characteristic functions .
Now P is the closest element to from Y, hence is approximable by A0 g, with A0 some linear
combination of the generators W .j Þ; kþ/ of A, and A0 g is in turn approximable by A0 h with any
h ∈ L2 sufficiently close to g. We conclude that, given ž > 0, there exists Ž such that if g − h < Ž,
then Ah, the closure of the subspace Ah, is not more than ž further from than is Y. If, moreover,
h ∈ Y, then Ah ⊂ Y and so cannot be nearer to than is Y. Thus Ph ≤ P ≤ Ph + ž.
By choosing Ž sufficiently small, we can similarly approximate each P that figures in the trace,
thereby approximating trA P arbitrarily well by trA Ph . If h ∈ Y , the same argument applies to
trA P .
Let u be a smooth function of compact support with g − uL2 < Ž; the operator Tu;1=þ;1=Þ is
then always bounded. Let h = P .P u/, which exhibits that h ∈ Y; and since P and P commute,
then also h = P .P u/ ∈ Y . Moreover, since g ∈ Y ∩ Y , g = P P g, and
g − h = P P .g − u/ ≤ g − u < Ž :
465
Gabor Time-Frequency Lattices
It remains to show that Th;Þ;þ is bounded. Invoking (6.14) and using the fact that P ∈ A , P ∈ A
in (6.12), we have
∞ > Tu;1=þ;1=Þ ≥ Tu;1=þ;1=Þ P = TP u;1=þ;1=Þ =
p
p
p
Þþ TP u;Þ;þ ≥ Þþ TP u;Þ;þ P = Þþ Th;Þþ ;
showing that Th;Þ;þ is bounded. This completes the proof of Theorem 6.2.
2
Appendix 6.1
Here we show that A is generated by W . mþ ; Þn /. A contains the algebra generated by W .j Þ; 0/,
which includes multiplication by all bounded functions of period 1=Þ; an operator
T ∈ A then
P
commutes with multiplication by such functions. Choosing as an instance k − .t + Þk /, with
− .t/ the characteristic function of an arbitrarily small interval centered at − , shows that for smooth
functions f .t/ of compact support, the value of Tf at a point t = − is a bounded linear functional
of {f .− − Þk /}, −∞ < k < ∞. Thus
Tf =
X
k
k
mk .t/f t −
;
Þ
(A1.1)
P
and k |mk .t/|2 is bounded uniformly in t since T is bounded in L2 . Now the commutativity of T
and W .0; þ/ means that [Tf ].t + þ/ = T [f .t + þ/]; whence by (A1.1) each mk .t/ has period þ.
Analogously, letting C denote the von Neumann algebra generated by W . mþ ; Þn /, each S ∈ C has the
form
Sf =
X
nj .t/f .t − jþ/ ;
(A1.2)
j
P
with nj .t/ of period Þ1 , and |nj .t/|2 uniformly bounded in t.
We want to show that S and T commute on a dense set in L2 .R/. To that end, suppose that
the support of f is smaller than min[þ; Þ1 ], whereupon the components in (A1.1) or (A1.2) do not
overlap. Further, let nj.M/ be smooth, have period Þ1 , and approximate nj in L2 [0; Þ1 ] well enough so
that
lim
M→∞
X
nj − nj.M/ 2L2 [0;1=Þ] = 0 :
j
Then
Sf =
lim
M;N →∞
X
|j |≤N
nj.M/ .t/f .t − jþ/
466
I. Daubechies, H. J. Landau, and Z. Landau
in L2 .R/, with the functions on the right smooth and of compact support and, hence, in the domain
of (A1.1). Since T is bounded,
T Sf =
=
=
X
lim
M;N →∞
X X
lim
M;N →∞
XX
j
|j |≤N
T nj.M/ .t/f .t − jþ/
k
|j |≤N
mk .t/nj.M/
k
mk .t/nj .t/f
k
t−
Þ
k
f t − jþ −
Þ
k
;
t − jþ −
Þ
the sum here likewise consisting of nonoverlapping components. By the analogous argument with
the roles of S and T reversed, STf has the same value. Thus T S = ST on the space of all linear
combinations of smooth functions of small support, which is dense in L2 .R/; whence the operators,
being bounded, commute everywhere. We conclude that C commutes with A , that is, C ⊂ .A / .
However, C ⊂ A ; whence .A / ⊂ C . Thus by the double commutant theorem
C = .A / = A
or equivalently
C = A ;
as was to be shown.
Appendix 6.2
The G–N–S construction realizes A on a Hilbert space L2 .A / that is naturally identifiable with
` ; moreover, the operators of A act simply on L2 .A / and can be carried easily to their counterparts
on `2 . Specifically, let V denote the unitary map from an element in L2 .A / to its coefficients in the
basis (6.10). We have seen in (6.13) that
2
V Tf∗;Þ;þ Tg;Þ;þ = .Þþ/−1 Tg;1=þ;1=Þ f :
(A2.1)
We can use (A2.1) to carry an operatorPon L2 .A / into its correspondent on `2 . P
To this end, we note
that if X ∈ L2 .A / has the expansion j;k W . þj ; Þk /, then X ∗ corresponds to j;k W ∗ . þj ; Þk / =
P
j;k e2³ i j k=Þþ W .− Þj ; − þk /. Thus
V X∗ = U V X ;
(A2.2)
467
Gabor Time-Frequency Lattices
with
U j;k ≡ −j;−k e−2³ i j k=Þþ ;
hence
U ∗ = U −1 = U :
Moreover, as an identity of sequences, and regardless of their square-summability,
U Tp;1=þ;1=Þ q = U
=
²−
q; W
²−
p; W
j k
;
þ Þ
j k
;
þ Þ
)
צ (−
×
j
k
−2³
i
j
k=Þþ
p = q; W − ; −
p e
þ
Þ
צ
q = Tq;1=þ;1=Þ p :
(A2.3)
(U is the R of (3.2) combined with the complex conjugation there.) Thus if X ∈ A is a linear
combination of a finite number of W . þj ; Þk /, then by (6.12), (A2.1), (A2.3), and the identity Xg ≡
∗
Tg;1=þ;1=Þ
V X,
V Tf∗;Þ;þ Tg;Þ;þ X = V Tf∗;Þ;þ TX∗ g;Þ;þ = .Þþ/−1 TX∗ g;1=þ;1=Þ f
= .Þþ/−1 U Tf ;1=þ;1=Þ X∗ g
∗
= .Þþ/−1 U Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
V X∗
∗
= .Þþ/−1 U Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
UVX:
(A2.4)
Since Tf∗;Þ;þ Tg;Þ;þ is bounded (on L2 , hence by the G–N–S correspondence also on L2 .A /), (A2.4)
can be extended by continuity to all X ∈ L2 .A /; whence
∗
Tf∗;Þ;þ Tg;Þ;þ = V ∗ U .Þþ/−1 Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
UV
on L2 .A / ;
∗
U V Tf∗;Þ;þ Tg;Þ;þ V ∗ U = .Þþ/−1 Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
(A2.5)
on `2 :
∗
Thus Tf∗;Þ;þ Tg;Þ;þ , viewed as acting on L2 .A /, is unitarily equivalent to .Þþ/−1 Tf ;1=þ;1=Þ Tg;1=þ;1=Þ
2
on ` , except for a complex conjugation, U being conjugate-linear.
468
I. Daubechies, H. J. Landau, and Z. Landau
Appendix 6.3
P
Here we prove that the convergence of j;k W . þj ; Þk / to Tf∗;Þ;þ Tg;Þ;þ in the norm of L2 .A /,
P
asserted by (6.10), implies that, for h ∈ L2 , j;k W . þj ; Þk / h converges to Tf∗;Þ;þ Tg;Þ;þ h in L2 . We
P
begin by showing that j;k W . þj ; Þk / converges strongly on functions u of compact support. For
since Tf∗;Þ;þ Tg;Þ;þ ∈ A , there exist linear combinations of the generators of A ,
Sn ≡
X
|j;k|≤n
.n/
aj;k
W
j k
;
þ Þ
which converge to it strongly and, hence, as we have seen, also in L2 .A /. Consequently,
X
.n/
|aj;k
− j;k |2 = Sn − Tf∗;Þ;þ Tg;Þ;þ 2L2 .A / = žn2 → 0 ;
P
.n/
with aj;k
= 0 outside |j; k| ≤ n. Set Tn =
Sn u = Tn u +
X
.n/
.aj;k
|j;k|≤n
− j;k / W
|j;k|≤n j;k
j k
;
þ Þ
(A3.1)
W . þj ; Þk /. Then
.n/
∗
u = Tn u + Tu;1=þ;1=Þ
{aj;k
− j;k }|j;k|≤n : (A3.2)
∗
As u is compactly supported, Tu;1=þ;1=Þ
is bounded by Proposition 2.1, and since by (A3.1)
X
|j;k|≤n
.n/
|aj;k
− j;k |2 ≤
X
.n/
|aj;k
− j;k |2 = žn2 ;
the last term in (A3.2) converges to 0. Thus from (A3.2)
lim Tn u = lim Sn u = Tf∗;Þ;þ Tg;Þ;þ u ;
n→∞
n→∞
the last equality by strong convergence of Sn . By (6.13), we recognize the left-hand limit to be
∗
Tg;1=þ;1=Þ f , and so find that, as elements of L2 ,
.Þþ/−1 Tu;1=þ;1=Þ
∗
Tf∗;Þ;þ Tg;Þ;þ u = .Þþ/−1 Tu;1=þ;1=Þ
Tg;1=þ;1=Þ f:
(A3.3)
For arbitrary h ∈ L2 , let un → h in L2 , with each un of compact support. Applying (A3.3), since
Tf∗;Þ;þ Tg;Þ;þ is bounded,
Tf∗;Þ;þ Tg;Þ;þ h = lim .Þþ/−1 Tu∗n ;1=þ;1=Þ Tg;1=þ;1=Þ f :
n→∞
(A3.4)
Gabor Time-Frequency Lattices
469
We now observe that the right-hand operator is closed as a map of un . For if ∈ `2 is a fixed
sequence, ², ¦ are constants, and Tu∗n ;²;¦ → q ∈ L2 , then with ' a function of compact support,
.q; '/ = lim .Tu∗n ;²;¦ ; '/ = lim . ; Tun ;²;¦ '/ :
n→∞
n→∞
(A3.5)
Again by Proposition 2.1, T';²;¦
is bounded, and therefore T';²;¦ un → T';²¦ h. But since T';²;¦ un
¯
and Tun ;²;¦ ' are unitarily equivalent, there is corresponding convergence in (A3.5). Consequently,
.q; '/ = . ; Th;²;¦ '/ ;
∗
∗
whence is in the domain of Th;²;¦
, with q = Th;²;¦
. From (A3.4) we thus obtain
∗
Tg;1=þ;1=Þ f ;
Tf∗;Þ;þ Tg;Þ;þ h = .Þþ/−1 Th;1=þ;1=Þ
as was to be shown.
7. Dual Functions That Optimize Other Norms
As remarked by Wexler and Raz in [5], there are cases when other functions than the minimal
L2 -dual function g # = g̃ for the frame .gmÞ;nþ /m;n∈Z lead to better concentration and/or convergence.
This can easily be illustrated by the following example. If one takes the limit in which both Þ and þ
tend to zero, then the discrete family gmÞ;nþ becomes the family gp;q = W .p; q/g, where p; q range
continuously over R. It is then well known (see, for example, [10]; a short review is also in [3]) that,
for any f; h in L2 .R/,
Z Z
f; hp;q gp;q dp dq = g; hf :
R2
This is exactly of the same form as the expansions in the gmÞ;nþ we saw previously, with an integral
over p; q replacing the sum over m; n. It follows that all h in L2 .R/ for which g; h = 1 are
dual functions for the gp;q in the sense that the above integral reproduces each f exactly. Among
these, the particular choice h = g−2 g is the one with minimal L2 -norm. One can easily imagine,
however, if either g itself or its Fourier transform ĝ is rather spread out, that other, more concentrated
choices of h might be preferable. For instance, if g = √12 H0 + √12 Hk , where Hn denotes the nth
Hermite function, then theR dual function with
L2 -norm is g itself, but the dual function G
R minimal
2
2
2
concentrated in both
that minimizes |||G||| =
P |xG.x/| dx + |¾ Ĝ.¾2/| d¾Pis∞different and more
2
x and ¾ . Writing G = √
Minimizing this
n ¼n Hn , one finds |||G||| =
n=0 .n + 1/|¼n | . √
√ under
1
,
¼
=
2 k+2
.
the constraint ¼0 + ¼k = 2 leads to ¼n = 0 if n = 0, n = k and to ¼0 = 2 k+1
k
k+2
Figure 1 shows a graph of both g and G for k = 10; G is clearly more concentrated and less oscillating than g. A different example, still in the limit case of vanishing Þ; þ, is given by the choice
470
I. Daubechies, H. J. Landau, and Z. Landau
(a)
0.5
0
–10
–5
0
√1
2
H0 +
√1
2
5
10
5
10
H10
(b)
0.6
0.4
0.2
0
–10
–5
0
√ 11
2 12 H0 +
1
12
H10
Ð
FIGURE 1
Graphs of (a) g.x/ =
minimal.
√1 [H0 .x/ + H10 .x/];
2
(b) G.x/, the function dual to g for which
R
|xG.x/|2 dx +
R
|¾ Ĝ.¾ /|2 d¾ is
Gabor Time-Frequency Lattices
471
g.x/ = ³ −1 e−x for x ≥ 0, g.x/ = 0 for x < 0, that is, the one-sided exponential. This function has a discontinuity, which leads to extensive spreading in the frequencyRdomain. One can find
dual functions that are much smoother;ð the dual function G that minimizes
.1 + |¾ |2 /|Ĝ.¾ /|2 d¾ ,
Ł
1
−|x|
−x
for instance, is given by G.x/ = 2 e
+ x.1 + sign.x//e ; both g and G are plotted in
Figure 2.
A similar thing happens for the discrete families gmÞ;nþ , although the analysis is a little less
straightforward. Assume that the gmÞ;nþ constitute a frame and that we are interested in finding the
dual function g b for which |||g b ||| = 3g b is minimal. (In the examples above, 3 = .−d 2 =dx 2 +
x 2 /1=2 and 3 = .−d 2 =dx 2 + 1/1=2 , respectively.) Because of Proposition 3.8, this is equivalent to
finding the function G with smallest L2 -norm that satisfies
Tg;1=þ;1=Þ 3−1 G = Þþ e0;0
(7.1)
and then taking g b = 3−1 G. Note that we are implicitly assuming that 3 has a bounded inverse
3−1 . This is not a severe restriction: we can assume 3 ≥ 0 without loss of generality (otherwise,
replace 3 by .3∗ 3/1=2 ), and we can then add 1 to 3, if necessary, without changing the nature of the
smoothness or decay constraint. We shall systematically assume 3 ≥ Id in what follows. Finding
the minimal L2 -solution G to (7.1) amounts to replacing Tg;1=þ;1=Þ by Tg;1=þ;1=Þ 3−1 in the argument
that led to the minimal norm solution g # in §4. By this argument, we are therefore led to the choice
∗
∗
Þþ 3−1 Tg;1=þ;1=Þ
.Tg;1=þ;1=Þ 3−2 Tg;1=þ;1=Þ
/−1 e0;0
for G. It is, however, not clear in what sense this should be understood, since the operator
∗
Tg;1=þ;1=Þ 3−2 Tg;1=þ;1=Þ
, although well defined and bounded on `2 .Z2 /, is generally not invertible
∗
is. The solution is to introduce an extra operator on `2 .Z2 /, typically
even if Tg;1=þ;1=Þ Tg;1=þ;1=Þ
also unbounded, which, in a sense that will become more precise below, complements on `2 .Z2 / the
action of 3 on L2 .R/. Concretely, we have
7.1.
Proposition
Assume that the gmÞ;nþ constitute a frame. Let 3; be .unbounded/ operators on L2 .R/,
` .Z /, such that 3; ≥ Id on their domains. Suppose that e0;0 lies in the domain of and that
there exist 0 < A ≤ B < ∞ so that, for all c in `2 .Z2 /,
2
2
∗
A −1 c2 ≤ 3−1 Tg;1=þ;1=Þ
c2 ≤ B −1 c2 :
(7.2)
Then the function g̃ 3 defined by
Ð−1
∗
∗
g̃ 3 = Þþ 3−2 Tg;1=þ;1=Þ
Tg;1=þ;1=Þ 3−2 Tg;1=þ;1=Þ
e0;0
is a dual function for the frame gmÞ;nþ ; moreover, for any other dual function g b , we have
3g̃ 3 ≤ 3g b :
(7.3)
472
I. Daubechies, H. J. Landau, and Z. Landau
(a)
1.5
1
0.5
0
–5
0
²
(b)
e−x
0
5
x≥0
x<0
0.8
0.6
0.4
0.2
0
–5
0
8
<
:
x≤0
2
e−|x|
2
e−x
5
+ xe−x
x≥0
FIGURE 2
Graphs of (a) g.x/ = e−x [0;∞/ .x/; (b) G.x/, the function dual to g for which
R
[1 + |¾ |2 ] |Ĝ.¾ /|2 d¾ is minimal.
Gabor Time-Frequency Lattices
473
∗
First of all, note that g̃ 3 is well defined, since Tg;1=þ;1=Þ 3−2 Tg;1=þ;1=Þ
is invertible
by (7.2). Next, Tg;1=þ;1=Þ g̃ 3 is obviously in the domain of , and Tg;1=þ;1=Þ g̃ 3 = Þþ e0;0 . Since
is invertible, it follows that Tg;1=þ;1=Þ g̃ 3 = Þþ e0;0 , so that g̃ 3 is a dual function for the frame
gmÞ;nþ . Finally, among all the functions that satisfy
Proof.
Tg;1=þ;1=Þ 3−1 3g b = Þþ e0;0 ;
the one for which 3g b is minimal is given by projection onto [Ker .Tg;1=þ;1=Þ 3−1 /]⊥ , which
leads to (7.3).
2
This reduces the problem to identifying, for a given 3, the appropriate operator and proving
bounds of type (7.2). We shall show here how this can be done for the two special choices 3 =
.1+x 2 /p and 3 = .−d 2 =dx 2 +1/q , which correspond to trying to find more localized g b or smoother
g b , respectively. In fact, we only need to analyze the first case since the second can be obtained
from the first by Fourier transform. Let us assume, therefore, that 3 is a simple multiplication
operator, .3f /.x/ = ½.x/f .x/. We shall likewise restrict our attention to operators of the form
.c/m;n = !n cm;n . The following proposition leads to estimates for A ; B .
7.2.
Proposition
Let 3; be as above. Define
S=
sup
n∈Z;x∈R
I=
R=
inf
n∈Z;x∈R
sup
þ n þþ2
þ
!n2 ½.x + `þ/−2 þg x + `þ −
þ ;
Þ
`∈Z
X
þ n þþ2
þ
!n2 ½.x + `þ/−2 þg x + `þ −
þ ;
Þ
`∈Z
X
X
n∈Z;x∈R k∈Z
k=0
þ
þ
þX
−2 þ
n
k
þ
þ
½ x + + `þ
!n !n+k þ
g.x + `þ/ g x + `þ −
þ :
þ `
þ
Þ
Þ
If S; R < ∞ and I > R, then (7.2) holds with A = .I − R/þ and B = .S + R/þ.
Proof.
Renaming −1 c = d, we need to derive bounds on
Z
∞
½.x/
−2
−∞
Define dn .x/ = þ −1=2
þ
þ2
þX
þ
þ
þ
!n dm;n gm=þ ; n=Þ .x/þ dx :
þ
þ m;n
þ
(7.4)
h
i
m
d
exp
−2³i
x
; we have
m∈Z m;n
þ
P
Z
0
þ
|dn .x/|2 dx =
X
m;n
|dm;n |2 :
(7.5)
474
I. Daubechies, H. J. Landau, and Z. Landau
It follows that
Z
(7.4) =
∞
þ½.x/
−2
−∞
=þ
XZ
`∈Z
=þ
þ
þ2
þX
þ
n
þ
−1 þ
!n dn .x/g x + `þ −
½.x + `þ/ þ dx
þ
þ n
þ
Þ
þ
0
Z
XX
(
=þ
þ
!n ! m
0
n;m
`
þ
þ2
þX
n þþ
þ
d .x/!n g x −
þ
þ dx
þ n n
Þ þ
XZ
n m
½.x + `þ/−2 dx
dn .x/ dm .x/ g x + `þ −
g x + `þ −
Þ
Þ
"
þ
|dn .x/|
2
0
n
X
`
!n2
þ n þþ2
þ
þg x + `þ −
þ ½.x + `þ/−2
Þ
#
)
dx + Rest
(7.6)
with
|Rest| ≤
X
Z
þ
!n !m
|dn .x/| |dm .x/| ×
0
n;m
n=m
þ
þ
þX þ
n m −2
þ
þ
g x + `þ −
g x + `þ −
½ .x + `þ/þ dx
þ
þ `
þ
Þ
Þ
0
Z
BX
≤@
!n !m
þ
0
n;m
n=m
0
Z
BX
!n !m
@
0
n;m
n=m
=
X
Z
!n !m
0
n;m
n=m
=
XZ
n
=
n
þ
0
þ
þ
11=2
þ
þ
þX þ
n
m −2
þ
þ C
|dm .x/|2 þ
g x + `þ −
g x + `þ −
½ .x + `þ/þ dx A
þ `
þ
Þ
Þ
þ
þ
þ
þX n m −2
þ
þ
|dn .x/| þ
g x + `þ −
g x + `þ −
½ .x + `þ/þ dx
þ
þ `
Þ
Þ
2
þ
þ
þX þ
k
n
n
þ
þ
−2
½ .x + `þ/þ dx
|dn .x/|
!n !n+k þ
g x + `þ −
g x + `þ − −
þ
þ
Þ
Þ
Þ
`
k=0
2
0
XZ
þ
11=2
þ
þ
þX þ
n
m −2
þ
þ C
|dn .x/|2 þ
g x + `þ −
g x + `þ −
×
½ .x + `þ/þ dx A
þ `
þ
Þ
Þ
X
þ
þ
þX
k
n −2 þþ
þ
|dn .x/|
!n !n+k þ
g.x + `þ/ g x + `þ −
½ x + `þ +
þ dx :
þ `
þ
Þ
Þ
k=0
2
X
(7.7)
475
Gabor Time-Frequency Lattices
Combining (7.5), (7.6), and (7.7) immediately leads to
þ.I − R/
X
|dm;n |2 ≤ .7:3/ ≤ þ.S + R/
X
2
|dm;n |2 :
m;n
m;n
We have tried out this approach on a function h similar to the one-sided exponential of Figure 2a
in order to find a dual function H with better frequency localization than h itself. The Fourier
transform of the one-sided exponential is .1 + i!/−1 ; in order to regularize its behavior at ∞, we
multiply this with a very wide Gaussian, defining
ĥ.!/ = .1 + i!/
−1
!2
exp −
50
or
h.x/ = .5e−25x
2
=2
∗ e/.x/ ;
where e.x/ = exp.−x/[0;∞/ . (In this section, we shall systematically normalize the Fourier transR
form as fˆ.¾ / = √12³ e−i¾ x f .x/ dx.) We wish to consider the functions hmÞ;nþ .x/ = e−2³ max
h.x − nb/ and to find a dual function H with better frequency concentration. Translating all this into
the Fourier domain and writing g.x/ = ĥ.x/, we find that the lattice of interest is given by gmÞ;nþ .x/,
b
with þ = 2³a, Þ = 2³
, g.x/ = .1 + ix/−1 exp.−x 2 =50/. We choose ½.x/ = .1 + x 2 /1=4 , and we
define the !n by
"
!n2
= inf
x∈R
X
½.x + `þ/
−2
`∈Z
þ n þþ2
þ
þg x + `þ −
þ
Þ
#−1
:
It follows that I = 1 (where I is defined as in Proposition 7.2). For a = b = 0:25, numerical
computation led to S 1:0803 and R :0397, so BA 1:16. To determine g̃ 3 , we then follow
the prescription (7.3). First, note that e0;0 = !0 e0;0 . Next, we compute the matrix for L =
2
−2 ∗
Tg;1=þ;1=Þ :
Id − A +B
Tg;1=þ;1=Þ 3
.Lc/m;n =
X
Lm;n;k;` ck;`
k;`
with
Lm;n;k;` = Žm;k Žn;` −
A
Þ
2
!n !` 3−2 g þk ;`=Þ ; gm=þ ; n=Þ :
+B
In practice, for our example, very few of these matrix elements are significant, except on or near the
diagonal m = k, n = `. This makes it easy to compute the iterates Ln e0;0 ; we then have
∗
.Tg;1=þ;1=Þ 3−2 Tg;1=þ;1=Þ
/−1 e0;0 = !0 d = !0 lim dN
N →∞
476
I. Daubechies, H. J. Landau, and Z. Landau
where d0 =
e0;0 and dN is defined recursively by dN = d0 + LdN −1 . As in [3], the limit
N ½
.
converges exponentially fast; the error made by truncating at step N is of the order O BA − 1
Since
2
A +B B
A
− 1 = 0:16 in our case, a few iterations suffice to obtain good accuracy.
Next, we use the dm;n to define the function
X
ð ∗
Ł
Þþ Tg;1=þ;1=Þ
d .x/ = Þþ
!n dm;n gm=þ;n=Þ .x/ :
m;n
Finally, we multiply this by .1 + x 2 /−1=2 = ½.x/−2 to obtain g̃ 3 . The inverse Fourier transform of
g̃ 3 is then the function H dual to the hma;nb (with a = b = 0:25) that minimizes .1 − 1/1=2 H; H .
Rather than computing this inverse Fourier transform numerically from g̃ 3 , we can use the integral
representation
2 −1=2
.1 + /
Z
2
=√
³
∞
e−.1+
2
/s 2
ds
0
to write
H .t/ =
"
d2
1− 2
dt
1
=√
³
Z
∞
0
−1=2 #
F .t/
1 −1=s 2
e
s
Z
∞
e− 4 s
1 2
.t−t /2
F .t / dt ds
−∞
where
F .t/ = Þþ
X
Ð∨
!n dm;n gm=þ;n=Þ .t/
m;n
= Þþ
X
m;n
!n dm;n e
i n=Þ.x−2³ m=þ/
m
h x − 2³
þ
:
Figure 3 illustrates this example. Figures 3a and 3b show h and the absolute value of its Fourier
transform, |g|; Figures 3c and 3d plot the dual function H and the absolute value of its Fourier
transform, |g̃ 3 |. For comparison, Figures 3e and 3f show the dual function h̃ = h# with minimal L2 -norm and the absolute value of its Fourier transform; clearly h̃ has sharper transitions
than H .
477
Gabor Time-Frequency Lattices
(a)
(b)
1
1.5
1
0.5
0.5
0
0
–5
0
5
(c)
–10
–5
0
5
10
–10
–5
0
5
10
–10
–5
0
5
10
(d)
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.1
0
0
–5
0
5
(e)
(f)
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
–5
0
5
FIGURE 3
(a) The function h and (b) the absolute value |ĥ| of its Fourier transform; (c) .Þþ/−1 H and (d) .Þþ/−1 |Ĥ |, where H is
the dual function that minimizes .1 − 1/1=4 H ; (e) .Þþ/−1 h̃ and (f) .Þþ/−1 |b̃
h|, where h̃ = h# is the standard,
L2 -minimal dual function.
478
I. Daubechies, H. J. Landau, and Z. Landau
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Received December 15, 1994
(I. Daubechies and H. J. Landau) AT&T Bell Laboratories, Murray Hill, New Jersey 07974
(Z. Landau) Department of Mathematics, University of California, Berkeley, California 94720