Gabor Frame Expansions
in Besov Spaces
René Koch
[email protected]
Introduction
Mathematical Preparations
We consider the sets Q0 := [−2, 2] and
j−1
j+1
Qj := x ∈ R | 2
≤ |x| ≤ 2
for j ∈ N.
Definition For J ⊆ N0 we define the set
This family of sets fulfills the conditions
[
Qj = R,
i)
j∈N0
ii) Qi ∩ Qj = ∅ for |i − j| > 2.
∗
J := {i ∈ N0 : ∃j ∈ J such that Qi ∩ Qj 6= ∅}
(k−1)∗ ∗
and inductively j (k)∗ = j
with j (0)∗ = {j}. Moreover,
:= i∈j (k)∗ ηi .
Lemma 1 For each l ∈ N exists A > 0 such that
1/q

1/q

∞
∞
q
q
X
X

(l)∗ ˆ 
jsq jsq ˆ


= A · kf kBqs .
≤A·
2 ηj f ˇ 2
2 ηj f ˇ 2
L
j=0
Next, we pick a nonnegative smooth function
η0 ∈ Cc∞ (R) such that supp η0 ⊆ Q0 as well
as η0 ≡ 1 on [−1, 1] and define the family of
functions
−j
−j+1
ηj := η0 2 x − η0 2
x
for j ∈ N.
(k)∗
ηj
P
L
j=0
Lemma 2 The following implication is true: [n − 2, n + 2] ∩ Qj 6= ∅ =⇒
Lemma 3 For every ε > 0 and f ∈ Bqs exists g ∈ S with kf − gkBqs < ε.
(3)∗
ηj
≡ 1 on [n − 2, n + 2].
Main Lemma
We want to show the existence of a constant K > 0 such that for every f ∈ Bqs , j ∈ N0 and arbitrary
finite set F ⊆ Z2 the following inequality is fulfilled
Thus, we obtain
i) supp ηj ⊆ Qj for j ∈ N0 ,
ii)
∞
X

 q
q
X
ˆ (3)∗ 

hf, Tk/2 Mn ψiTk/2 Mn ψ ηj ≤ K f ηj 2 .
L
2
(n,k)∈F
ηj (x) = 1 for x ∈ R,
j=0
(1)
L
iii) kηˇj kL1 < ∞ for j ∈ N0
Additionally, we introduce the discrete weight
N0 3 j 7→ 2js for s ∈ R. Lastly, we set ([3])
kf kBqs

1/q
∞
q
X

jsq ˆ

:=
2 ηj f ˇ
L2
j=0
This and Lemma 1 subsequently lead to the inequality
X
hf,
T
M
ψiT
M
ψ
k/2 n
k/2 n (n,k)∈F
≤ C kf kBqs
Bqs
for a suitable C > 0, independent of f, F, j.
and
n
o
s
0
Bq := f ∈ S : kf kBqs < ∞ .
Proof of Main Lemma
We take a finite set F ⊆ Z2 and a nonnegative auxiliary function ϕ ∈ Cc∞ (R), 0 ≤ ϕ ≤ 1, with the
additional properties suppϕ ⊆ [−2, 2] and ϕ ≡ 1 on [−3/2, 3/2]. Hence, we observe the equality
Starting Point
Cc∞ (R),
For ψ ∈ S with ψ̂ ∈
supp ψ̂ ∈ [−1, 1]
and
2
X ξ 7→
ψ̂(ξ − k) ≡ 1
k∈Z
the Gabor system Tk/2 Mn ψ (n,k)∈Z2 is a tight
2
frame
in
L
with
frame
bound
2.
This
entails
P
2
2
hf,
T
M
ψi
=
2kf
k
k/2 n
(n,k)∈Z2
L2 and the
unconditional convergence of the series
X
hf, Tk/2 Mn ψiTk/2 Mn ψ
(n,k)∈Z2
2
to
2f
for
every
f
∈
L
.
Furthermore,
M−k/2 Tn ψ̂
is
also
a
frame
with
frame
2
(n,k)∈Z
bound 2 [2].
Goal
Our aim is to establish the unconditional
convergence
of
the
series
P
(n,k)∈Z2 hf, Tk/2 Mn ψiTk/2 Mn ψ to 2f with
respect to k · kBqs for every f ∈ Bqs .
References
[1] L. Borup, M. Nielsen, Frame Decomposition
of Decomposition Spaces, 2007
[2] K. Gröchenig, Foundations of TimeFrequency Analysis, 2001
[3] H. Triebel, Theory of function spaces II, 2010
hf, Tk/2 Mn ψi = hfˆ, M−k/2 Tn ψ̂i = hTn ϕfˆ, M−k/2 Tn ψ̂i
Through a duality argument, the Plancherel theorem and frame properties of M−k/2 Tn ψ̂
we obtain the inequality
X
hf,
T
M
ψiT
M
ψ
n
n
k/2
k/2
k:(n,k)∈F
ˆ
≤ C · Tn ϕf L2
for all n ∈ Z.
(n,k)∈Z2
(2)
L2
Secondly, we define for n ∈ Z and j ∈ N0 the condition
A(n, j) := [n − 1, n + 1] ∩ Qj 6= ∅ and ∃k : (n, k) ∈ F.
The relation between the supports of Tn ψ̂ and ηj as well as ηj ≤
(3)∗
ηj
imply that


X
X
(3∗)
ˆ

 ηj hT
ϕ
f
,
M
T
ψ̂iM
T
ψ̂
n
n
n
−k/2
−k/2
n:A(n,j) k:(n,k)∈F
(3)
L2
is an upper bound for the left hand side in (1). Now Lemma 2 results in
(3)∗
Tn ψ̂ηj
= Tn ψ̂ for n, j such
(3)∗
ηj ,
that A(n, j) is valid. With some further reflections about the supports of Tn ψ̂ and
we arrive
P
(3)∗
at the inequality n:A(n,j) Tn ϕ ≤ M ηj
for some M > 0, independent of j. By piecing together
the previous results and inequality (2), we can ultimately conclude the validity of (1).
Concluding the Proof
Significance and Outlook
We show the desired unconditional convergence
for every f ∈ S via splitting the sum in the
definition of k · kBqs in a finite part and a tail.
Then we proceed to carry this convergence over
to every f ∈ Bqs by means of Lemma 3 and the
inequality established in the main lemma.
It is commonly known that wavelet systems are
tailor-made for the expansion of elements in
Besov spaces, whereas Gabor systems are ideally
suited for modulation spaces. In this respect,
the results of this poster are somewhat surprising. It is important to note, that this only holds
if the inner norm is measured in the space L2 .