On the Feichtinger conjecture
Volume
1081-3810
On the Feichtinger conjecture
Pasc Gavruta
Follow this and additional works at: https://repository.uwyo.edu/ela Recommended Citation
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ON THE FEICHTINGER CONJECTURE∗
PASC GA˘ VRUT¸ A†
AMS subject classifications. 46C05, 42C15.
1. Introduction. There are many variations of the Feichtinger Conjecture, all
equivalent with the following:
Every Bessel sequence of unit vectors in a Hilbert space can be partitioned into
finitely many Riesz sequences.
For details on the Feichtinger Conjecture and the connection with other problems,
see [
2
], [
3
], [
4
], [
9
] and [
10
], and references in these papers.
We denote by H a Hilbert space and F = {fn}n∈N ⊂ H. We say that F is a
Bessel sequence if there exists B > 0 so that
∞
X |hx, fni|2 ≤ Bkxk2,
n=0
547
We say that F is a Riesz sequence (or Riesz basic sequence) if there are A, B > 0
such that
A X |ck|2 ≤ k X ckfkk ≤ B X |ck|2
for any finite sequence (ck). Riesz sequences are particular cases of frames (see [
5
]).
Let be I ⊂ N. If F is a Bessel sequences in H, then FI = {fn}n∈I is clearly also
a Bessel sequence in H.
In [
5
], O. Christensen, using Schur’s test, give conditions on a sequence {fn}n∞=0
to be a Bessel sequence, that it only involves inner products between the elements
{fn}n∞=0:
Proposition 1.1. [
5
] Let {fn}n∞=0 be a sequence in H and assume that there
exists a constant B > 0 such that
∞
X |hfj, fki| ≤ B, ∀ j ∈ N.
k=0
Then {fn}n∞=0 is a Bessel sequence with bound B.
We call this sequences Bessel-Schur sequences.
The intrinsically localized sequences, introduced by K. Gr¨ochenig in [
8
], are
particular cases of Bessel-Schur sequences. In the same paper, he proves that every localized
frame is a finite union of Riesz sequences. Another type of localized sequences was
introduced by R. Balan, P.G. Casazza, C. Heil, and Z. Landau in [
1
]. They show that
the Feichtinger Conjecture is true for l1-self-localized frames which are norm-bounded
below. l1-self-localized Bessel sequences are also Bessel-Schur sequences.
On the other hand, we recall the following definition:
Definition 1.2. [
4
] A sequence {fn}n∈I of unit vectors in H is called separated
if there exists a constant γ < 1 such that
|hfn, fki| ≤ γ
for any n, k ∈ N, n 6= k.
In [
4
], the authors, among others, give the following result:
Theorem 1.3. Let H be a Hilbert space and let {fn}n∈I be a Bessel sequence
of unit vectors in H. Then {fn}n∈I can be partitioned into finitely many separated
Bessel sequences.
In the following, we prove that the Bessel-Schur sequences satisfies the Feichtinger
Conjecture. Also, we prove that every Bessel sequence of unit vectors in a Hilbert
space can be partitioned into finitely many uniformly separated sequences.
http://math.technion.ac.il/iic/ela
548
to be a Riesz sequence.
suppose that
Then, FI is a Riesz sequence.
bounded:
2. The results. First, we give a condition for a Bessel sequence of unit vectors
Theorem 2.1. Let FI = {fn}n∈I be a Bessel sequence of unit vectors. We
Proof. If FI is a Bessel sequence in H, then the following operators are linear and
2
T : l (I) → H, T (ci) = X cifi (synthesis operator),
Θ : H → l2(I),
Θx = {hx, fii}i∈I (analysis operator).
Moreover, Θ is the adjoint of T (see [
5
]).
For c = (cn)n∈I ∈ l2(I), we have
and hence,
(ΘT )(c) = hX ckfk, fji
=
X ckhfk, fji
j∈I
j∈I
i∈I
k∈I
k∈I
(ΘT )(c) − c =
X ckhfk, fji
k6=j
j∈I
.
By Cauchy-Schwartz inequality, it follows
2
k(ΘT )(c) − ck2 = X
X ckhfk, fji
2
j∈I k6=j
j∈I k∈I
k6=j
j∈I k∈I
k6=j
j∈I k∈I
k6=j
≤ X
X |ck||hfk, fji|1/2 · |hfk, fji|1/2
≤ X
2
X |ck| |hfk, fji|
≤ σ X
2
X |ck| |hfk, fji| .
X |hfk, fji|
k∈I
k6=j
2
On the Feichtinger Conjecture
549
Changing the order of summation, we obtain
k(ΘT )(c) − ck22 ≤ σ X |ck|2 X |hfk, fji|
k∈I j∈I
j6=k
≤ σ2kck22,
and so, kΘT − Ik ≤ σ < 1.
Therefore, ΘT is invertible, thus Θ is surjective. It follows that FI is a
RieszFischer sequence. From Theorem 3 in [13, Ch. 4, Sec. 2], we have that there exists
A > 0 so that
A X |ck|2 ≤ k X ckfkk2
k X ckfkk2 ≤ B X |ck|2
for every finite sequence (ck). Since FI is a Bessel sequence, we have
for (ck) finite sequence (see [
5
]). So, FI is a Riesz sequence.
Theorem 2.2. Every Bessel-Schur sequence of unit vectors is union of finite
Riesz sequences.
Proof. Let j ∈ N fixed. We have:
(2.1)
and hence,
We denote
We have aij = aji ≥ 0 and aii = 0.
The relation (2.1) is equivalent with
∞
X |hfj , fii| ≤ B,
i=0
X |hfj , fii| ≤ B − 1, for any j ∈ N.
i=0
i6=j
aij =
(|hfj , fii|,
0,
j 6= i,
j = i.
sup X aij ≤ B − 1.
j∈N i∈N
sup X aij ≤
j∈Ip i∈Ip
B − 1
;
2
By Mills’ Lemma (see [6, Ch. X] or [
12
]) there is a partition N = I1 ∪ I2 such that
http://math.technion.ac.il/iic/ela
550
By iteration, for any m ≥ 1, there is a partition N = I1 ∪ I2 ∪ . . . ∪ I2m such that
j∈Ip i∈Ip
sup X aij ≤
B − 1
2m
,
2m
3. An equivalent form of the Feichtinger conjecture. We consider the
following class of sequences.
Definition 3.1. Let FI = {fn}n∈I (...truncated)