Weaving Properties of Hilbert Space Frames
Peter G. Casazza
Richard G. Lynch
Director: The Frame Research Center
Department of Mathematics
University of Missouri
Columbia, MO, 65211
Website: http://www.framerc.org
Department of Mathematics
University of Missouri
Columbia, MO 65211
Email: rilynch37@gmail.com
Abstract—We will prove some new results in the theory of
Weaving Frames. Two frames {ϕi }i∈I and {ψi }i∈I in a Hilbert
space H are woven if there are constants 0 < A ≤ B so that for
every subset σ ⊂ I, the family {ϕi }i∈σ ∪ {ψi }i∈σc is a frame
for H with frame bounds A, B. We begin by introducing the
main results in weaving frames. We then prove some new basic
properties. This is followed by showing a fundamental connection
between frames and projections, providing intuition on woven
frames. Finally, a weaving equivalent of an unconditional basis
for weaving Riesz bases is considered.
real Hilbert space and I can represent a finite or countably
infinite index set.
I. I NTRODUCTION
where A and B are the lower Riesz bound and upper Riesz
bound, respectively.
This note focuses on an intriguing area of research called
weaving frames [1]. Two frames {ϕi }i∈I and {ψi }i∈I for a
Hilbert space H are woven if there are constants 0 < A ≤ B
so that for every subset σ ⊂ I, the family {ϕi }i∈σ ∪ {ψi }i∈σc
is a frame for H with frame bounds A and B.
A potential application of weaving frames together is dealing with wireless sensor networks which may be subjected
to distributed processing under different frames. The theory
could also have use in the preprocessing of signals using Gabor
frames.
In this paper, we review fundamental properties in weaving
frames from [1], followed by some new basic properties. A
relation of frames to projections is then considered and gives
a better understanding of what it really means for two frames
to be woven. We conclude by showing a weaving equivalent
of an unconditional basis.
We note that there is another concept in the literature which
uses multiple frames called Quilted Frames introduced by
M. Dörfler [4], [5], and are seemingly unrelated to woven
frames. Quilted Gabor frames are systems constructed from
globally defined frames by restricting these to certain, possibly
compact, regions in the time-frequency or time-scale plane.
II. F RAME T HEORY P RELIMINARIES
A brief introduction to frame theory is given in this section,
which contains the necessary background for this paper. For a
thorough approach to the basics of frame theory, see [2], [3].
Unless otherwise noted, H will denote either a finite or infinite
The authors were supported by NSF 1307685; NSF ATD 1042701; NSF
ATD 1321779; AFOSR DGE51: FA9550-11-1-0245.
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
Definition 1. A family of vectors Φ = {ϕi }i∈I in H is said
to be a Riesz basis if there are constants 0 < A ≤ B < ∞ so
that for all {ci }i∈I ∈ `2 (I),
2
X
X
X
2
ci ϕi |ci |2
A
|ci | ≤ ≤B
i∈I
i∈I
i∈I
Riesz bases have proved to be very useful in some applications in which the assumption of orthonormality is too
extreme. Similar to an orthonormal basis, Riesz bases satisfy
uniqueness of a decomposition as well as stability.
There are times when assuming the sequence is a Riesz basis
is even too strong. In these cases we work with frames, which
are redundant families of vectors that have proper subsets
spanning the space. Redundancy is the fundamental property
of frames which makes them so useful in practice.
Definition 2. A family of vectors Φ = {ϕi }i∈I in H is said
to be a frame if there are constants 0 < A ≤ B < ∞ so that
for all x ∈ H,
X
Akxk2 ≤
|hx, ϕi i|2 ≤ Bkxk2 ,
i∈I
where A and B are a chosen lower frame bound and upper
frame bound, respectively. If only B is assumed, then it is
called a B-Bessel sequence. If A = B, it is said to be an
A-tight frame and if A = B = 1, it is a Parseval frame.
The values {hx, ϕi i}i∈I are called the frame coefficients of
the vector x ∈ H with respect to the frame Φ.
If Φ = {ϕi }i∈I is a sequence in H, then the analysis
operator of Φ is the operator T : H → `2 (I) given by
T x := {hx, ϕi i}i∈I
and the associated synthesis operator is given by the adjoint
operator T ∗ : `2 (I) → H and satisfies
X
T ∗ {ci }i∈I :=
ci ϕi .
i∈I
The frame operator S : H → H is defined by S := T ∗ T and
satisfies
X
Sx = T ∗ T x =
hx, ϕi iϕi
i∈I
for any x ∈ H. These three operators are well-defined when
the sequence Φ is assumed to be at least a Bessel sequence.
If Φ is a frame with upper and lower bounds A and B,
respectively, then the frame operator is a positive, self-adjoint,
and invertible operator that also satisfies for any x ∈ H,
X
hAx, xi ≤ hSx, xi = kT xk2 =
|hx, ϕi i|2 ≤ hBx, xi,
i∈I
and hence operator inequality A · Id ≤ S ≤ B · Id holds.
Also, note that {S −1/2 ϕi }i∈I is a Parseval frame, called the
canonical Parseval frame of Φ. Finally, the norm of S is
kSk = kT ∗ T k = kT k2 .
III. W EAVING F RAMES P RELIMINARIES
This section is dedicated to a brief introduction to weaving
frames and is grounded in reviewing results from [1]. We begin
with the formal definition in full generality.
Definition 3. A finite family of frames {ϕij }M
j=1,i∈I in H is
said to be woven if there are universal constants A and B so
M
that for every partition {σj }M
j=1 of I, the family {ϕij }j=1,i∈σj
is a frame for H with lower and upper frame bounds A and B,
respectively. Each family {ϕij }M
j=1,i∈σj is called a weaving.
The overall goal is to discover conditions for which a family
of frames is woven. Proposition 3.1 from [1] shows one does
not need to check for a universal upper bound, as it is always
given by the sum of the upper frame bounds.
A natural follow-up question is whether one must check
for a universal lower bound. The answer is obviously no in
the finite dimensional case since there are only finitely many
ways to partition the index set and thus a universal bound is
easily obtained. However, in the infinite dimensional setting it
is not immediately clear that the lower frame bounds of the
weavings do not tend to zero. To show that a univeral bound
must be obtained, we introduce a weaker form of weaving.
{ϕij }M
j=1,i∈I
Definition 4. A family of frames
in H is said
to be weakly woven if for every partition {σj }M
j=1 of I, the
family {ϕij }M
is
a
frame
for
H.
j=1,i∈σj
One of the main theorems in [1] (Theorem 4.5) proves that
weakly woven is equivalent to the frames being woven.
Definition 6. If W1 and W2 are nontrivial subspaces of H,
define
dW1 (W2 ) := inf{kx − yk : x ∈ W1 , y ∈ SW2 }
where SW2 = SH ∩ W2 and SH is the unit sphere in H.
Similarly define dW2 (W1 ) by swapping the roles of W1 and
W2 . The distance between W1 and W2 is defined as
d(W1 , W2 ) := min{dW1 (W2 ), dW2 (W1 )}.
The following theorem, Theorem 5.7 of [1], gives an important relationship in weaving Riesz bases with this distance.
∞
Theorem 7. If Φ = {ϕi }∞
i=1 and Ψ = {ψi }i=1 are Riesz
bases in H, then the following are equivalent:
(i) Φ and Ψ are woven.
(ii) For any σ ⊂ N, d span{ϕi }i∈σ , span{ψi }i∈σc > 0.
(iii) There is a constant D > 0 so that for any σ ⊂ N,
d span{ϕi }i∈σ , span{ψi }i∈σc ≥ D.
This will later be generalized to frames by using projections
instead of distance.
Perturbations were also considered in [1]. Intuitively, a
frame and a small perturbation of itself should be woven as
the next proposition confirms.
Proposition 8. [1] If Φ = {ϕi }i∈I is a frame with bounds A
and B, and F is an invertible operator satisfying
A
,
B
are woven.
kId − F k2 <
then {ϕi }i∈I and {F ϕi }i∈I
This result extends easily to the case of finitely many
operators with minor adjustments.
Proposition 9. Let {ϕi }i∈I be a frame for H and {Fj }nj=1
invertible operators on H with
r
n
X
A
kId − Fj k <
.
B
j=1
Then {ϕi }i∈I and {Fj ϕi }nj=1,i∈I are woven.
To conclude this section, we note that the whole theory
of weaving frames was born out of the idea preprocessing
of signals using Gabor frames. However, we were unable
to answer the following problem, which still remains open.
Here Tam is the shift operator by a factor of am and Mbn is
modulation operator by a factor of bn.
∞
Theorem 5. Given two frames {ϕi }∞
i=1 and {ψi }i=1 for H
the following are equivalent:
(i) The two frames are woven.
(ii) The two frames are weakly woven.
Problem 10. Given a fixed lattice generated by a, b > 0 with
2
ab < 1 and rotated Gaussians Uj gαj , where gαi (x) = e−αi x ,
are the Gabor frames {Tam Mbn Uj gαj }M
j=1,m,n∈Z woven?
Therefore, it only needs to be checked that each weaving is
a frame, possibly each having different lower frame bounds.
Another direction is to study the special case when each
frame is a Riesz basis. It turns out that weaving Riesz bases
is classified by the following notion of distance.
In this section, we prove some new basic properties in
the theory of weaving frames. We begin by showing that
an invertible operator applied to woven frames leaves them
woven. However, applying an operator to only one of the
IV. W EAVING F RAMES
frames will not. We then show that multiplying the frame
vectors of two woven frames by uniformly bounded constants
gives woven frames.
Afterwards, we see if the weaving property may be checked
on smaller index sets and look at the repercussions of deleting
frame vectors. Finally, we prove a relationship between the
norms of the frame operators of the original frames and the
frame operators of the weavings.
Proposition 11. Suppose {ϕij }M
j=1,i∈I is a woven family of
frames for H with common frame bounds A and B. If F is an
invertible operator on H, then {F ϕij }M
j=1,i∈I is also woven
with bounds AkF −1 k−2 and BkF k2 . In particular, the bounds
do not change if F is unitary.
Proof. It is a known fact that if a frame has bounds A and
B, then applying an invertible operator F to it gives a frame
with bounds AkF −1 k−2 and BkF k2 . Since the sequence
{ϕij }M
j=1,i∈σj is a frame with lower and upper bounds A
and B, respectively, for any partition {σj }M
j=1 of I, then the
sequence {F ϕij }M
is
a
frame
with
bounds
AkF −1 k−2
j=1,i∈σj
2
M
and BkF k . That is, {F ϕij }j=1,i∈I is woven with universal
bounds AkF −1 k−2 and BkF k2 .
Remark 12. Proposition 11 can be relaxed to a bounded
operator F with closed range if F −1 is replaced with F † [3,
Prop 5.3.1].
The next result gives that a frame and a nonidentical
reordering of itself may not be woven.
Proposition 13. If {ϕi }i∈I is a Riesz basis with Riesz bounds
A, B and π is a permutation of I, then for every σ ⊂ I the
family {ϕi }i∈σ ∪{ϕπ(i) }i∈σc is a frame sequence with bounds
A and 2B. However, {ϕi }i∈I and {ϕπ(i) }i∈I are woven if and
only if π = Id.
Proof. For any x ∈ span {ϕi }i∈σ ∪ {ϕπ(i) }i∈σc and for any
σ ⊂ I,
X
X
X
|hx, ϕi i|2 +
|hx, ϕπ(i) i|2 ≥
|hx, ϕi i|2
i∈σ
i∈σ c
σ∪(σ c ∩π(σ c ))
2
≥ Akxk ,
since any subsequence of a Riesz basis is a Riesz sequence
with the same bounds. The upper frame bound is the sum of
the upper frames bounds, which is 2B. Note that it is not B
due to redundancy.
The however part is now proven via contraposition. Assume
π 6= Id so that π(i0 ) = j0 6= i0 for some i0 , j0 ∈ I. Let
σ = I\{i0 }. Then
{ϕi }i∈σ ∪ {ϕπ(i) }i∈σc = {ϕi }I\{i0 } ∪ {ϕj0 }
which is the set in which ϕj0 appears twice, but ϕi0 does not
appear at all and therefore the closure of the span is not the
entire space.
Since a frame is always woven with a copy of itself, the
previous proposition implies that applying an operator to only
one of the woven frames may not leave woven frames.
The following proposition gives conditions on multiplying
the frame vectors by individual constants and still be left with
woven frames.
Proposition 14. If {ϕi }i∈I and {ψi }i∈I are woven with universal lower and upper weaving bounds A and B, respectively,
and 0 < C ≤ |ai |2 , |bi |2 ≤ D < ∞ are constants, then
{ai ϕi }i∈I and {bi ψi }i∈I are also woven, but with lower and
upper bounds AC and BD, respectively. In particular,
1
1
√ ϕi
√ ψi
and
A
A
i∈I
i∈I
are woven with lower and upper bounds 1 and B/A, respectively.
Proof. A simple calculation yields this.
The next result gives that weavings may possibly be checked
on smaller index sets than the original.
Proposition 15. If J ⊂ I and {ϕi }i∈J and {ψi }i∈J are woven
frames then {ϕi }i∈I and {ψi }i∈I are woven.
Proof. For any σ ⊂ I and any x ∈ H we have
X
X
Akxk2 ≤
|hx, ϕi i|2 +
|hx, ψi i|2
i∈σ c ∩J
i∈σ∩J
≤
X
X
2
|hx, ϕi i| +
|hx, ψi i|2 .
i∈σ c
i∈σ
Since the upper bound is always given, this implies {ϕi }i∈I
and {ψi }i∈I are woven.
Hence, adding vectors {ϕ0j }j∈J and {ψj0 }j∈J (even letting
= 0 for all j ∈ J) to two woven frames still leaves two
woven frames. Therefore, asking for woven frames to have
some “property” for all subsets is not achievable.
It is possible to remove vectors from woven frames and still
be left with woven frames as the next result shows.
ϕ0j
Proposition 16. Suppose {ϕi }i∈I and {ψi }i∈I are woven with
universal constants A and B. If J ⊂ I and
X
|hx, ϕi i|2 ≤ Dkxk2
i∈J
for some 0 < D < A and for all x ∈ H, then {ϕi }i∈I\J and
{ψi }i∈I\J are also frames for H and are woven with universal
lower and upper frame bounds A − D and B, respectively.
Proof. The fact that B is an upper weaving bound is obvious.
Suppose that σ ⊂ I\J. Then for all x ∈ H,
X
X
|hx, ϕi i|2 +
|hx, ψi i|2
i∈σ
i∈(I\J)\σ
=
X
|hx, ϕi i|2 −
X
i∈σ∪J
+
X
i∈J
2
|hx, ψi i|
i∈(I\J)\σ
≥ (A − D)kxk2
|hx, ϕi i|2
so that a lower weaving bound is A − D. Taking σ = J c
and σ = ∅ gives that {ϕi }i∈I\J and {ψi }i∈I\J are frames,
respectively.
Proof. Let (Tj )σj be the analysis operator of Φj restricted to
the sum over σj . For any x ∈ H,
2
X
X
X
(Sj )σj x2 =
hx, ϕij iϕij An immediate corollary considers the case of when the
frame is woven with itself and hence vectors may possibly
be removed to be left with a frame.
j∈J
j∈J
j∈J
Corollary 17. If {ϕi }i∈I is a frame with lower frame bound
A and
X
|hx, ϕi i|2 ≤ Dkxk2
≤
The following example is an application of Proposition 16.
Example 18. Let {ei }∞
i=1 be an orthonormal basis for H.
Consider the two frames
100
100
e2 , . . . , e i ,
ei , . . .}
Φ = {ϕi }∞
i=1 = {e1 , 100e1 , e2 ,
2
i
Ψ = {ψi }∞
i=1 = {e1 , e1 , e2 , e2 , . . . , ei , ei , . . . , }.
These two frames are (1, 101)-woven, and it is clear that the
even terms in each frame may be removed to still leave two
woven frames. However, J = 2N does not satisfy Proposition
16. On the other hand, > 0 can be chosen so that
Φ = {ϕi, }∞
i=1
100
100
e2 , . . . , e i , ei , . . .}
2
i
and Ψ are still woven with lower woven bound one, but
X
|hx, ϕi, i|2 ≤ Dkxk2
= {e1 , 100e1 , e2 , i∈2N
for some 0 < D < 1 and thus the proposition applies showing
that the even terms are not necessary to determine weaving.
Finally, the next proposition gives a relationship between
the norms of the frame operators of the original frames and
the weavings.
Proposition 19. Let {Φj }j∈J be a (finite or infinite) collection
of frames Φj = {ϕij }i∈I for H with respective frame
operators Sj . Suppose there are constants A, B > 0 so that
for any partition {σj }j∈J of I, the collection
Ψ = {ϕij : i ∈ σj , j ∈ J}
is a frame for H with lower and upper frame bounds, A and
B, respectively. That is, the frames {Φj }j∈J0 are woven for
any finite J0 ⊂ J (by choosing σj = ∅ for all j ∈ J\J0 ). If
SΨ represents the frame operator of Ψ, then for any x ∈ H,
X
(Sj )σj x2 ≤ B SΨ 2 ,
A
j∈J
where (Sj )σj denotes the frame operator Sj with sum restricted to σj .
X
j∈J
B
X
|hx, ϕij i|2
i∈σj
= BhSΨ x, xi.
i∈J
for some 0 < D < A and for all x ∈ H, then {ϕi }i∈J c is a
frame with lower frame bound A − D.
i∈σj
X
(Tj∗ )σj (Tj )σj x2
=
Now using the inequality A·Id ≤ SΨ , where Id is the identity
operator, gives
X
(Sj )σj x2 ≤ BhSΨ x, xi = BhS 1/2 x, S 1/2 xi
Ψ
Ψ
j∈J
≤B
1
1/2
1/2
SΨ SΨ x, SΨ x
A
=
B
Sψ 2
A
as desired.
V. W EAVINGS AND P ROJECTIONS
This section is devoted to developing an important characterization of frames with projections. An immediate corollary
gives some intuition on what it means to weave two frames
together.
Theorem 20. Given vectors {ϕi }i∈I in H, the following are
equivalent:
(i) {ϕi }i∈I is a frame.
(ii) For any σ ⊂ I, let Wσ = span{ϕi }i∈σ , Wσc =
span{ϕi }i∈σc , and let P be the orthogonal projection of
H onto Wσ⊥ . Then P |Wσc is onto Wσ⊥ and {P ϕi }i∈σc
is a frame for Wσ⊥ .
Proof. The fact that (ii) ⇒ (i) is immediate by considering
σ = ∅. To prove (i) ⇒ (ii), note that {P ϕi }i∈I is a frame for
Wσ⊥ . However, P ϕi = 0 for all i ∈ σ and thus {P ϕi }i∈σc is
a frame for Wσ⊥ .
Let T1 : `2 (σ c ) 7→ Wσc and T : `2 (σ c ) 7→ Wσ⊥ be the
operators defined by T1 ei := ϕi and T ei := P ϕi for i ∈ σ c ,
where {ei }i∈σc is the standard orthonormal basis. It must be
checked that P |Wσc is onto. However, note that T is onto since
{P ϕi }i∈σc is a frame and that T = P |Wσc T1 . Hence, P |Wσc
must also be onto, concluding the proof.
Corollary 21. Given two frames Φ = {ϕi }i∈I and Ψ =
{ψi }i∈I for H, the following are equivalent:
(i) The frames Φ and Ψ are woven.
(ii) For any σ ⊂ I, let Wσ = span{ϕi }i∈σ , Wσc =
span{ψi }i∈σc , and let P be the orthogonal projection of
H onto Wσ⊥ . Then P |Wσc is onto Wσ⊥ and {P ϕi }i∈σc
is a frame for Wσ⊥ .
VI. U NCONDITIONAL C ONSTANTS OF W EAVING R IESZ
BASES
In this last section, we consider the weaving equivalent
of an unconditional basis for H. Recall that {xn }∞
n=1 is an
unconditional sequence in H if and only if there is a constant
C > 0 so that for all σ ⊂ HN and for all scalars {an }∞
n=1 we
have
X
X
X
X
∞
a
x
a
x
+
=
C
≤
C
a
x
a
x
i i
i i
i i
i i
i∈σ c
i∈σ
i∈σ
i=1
∞
Theorem 22. Let Φ = {ϕi }∞
i=1 and Ψ = {ψi }i=1 be Riesz
basic sequences in H with frame bounds A1 , B1 and A2 , B2
respectively. The following are equivalent:
(i) There are constants 0 < A ≤ B so that for every σ ⊂ N
the family {ϕi }i∈σ ∪ {ψi }i∈σc is a Riesz basis with
bounds A, B. That is, Φ and Ψ are woven.
(ii) There is a constant 0 < C satisfying for all scalars
{ai }∞
i=1 and all σ ⊂ N
Hence,
1 C
2C +1
!
X
X
ai ψi ai ϕi +
i∈σ c
i∈σ
(
)
X
X
C
≤
ai ψi max ai ϕi , C +1
i∈σ c
i∈σ
X
X
a i ϕi +
ai ψi ≤
.
i∈σ c
i∈σ
P
(iv) ⇒ (ii): If i∈σ ai ϕi = 0 we are done. So assume not
and by (iv) we have:
X
X
1
P
E≤
a
ϕ
+
a
ψ
i i
i i .
aϕ
i∈σ
i
i
i∈σ c
i∈σ
So (ii) holds.
At this point we know that (ii) ⇔ (iii) ⇔ (iv). Now (i) ⇒
(ii): Given σ and {ai }∞
i=1 , by (i) we have
2
X
X
2
ai ϕi |ai |2
≤ B
i∈σ
i∈σ
!
X
X
X
ai ϕi +
a i ψi ai ϕi ≤ C
.
(iii) There is a constant 0 < D satisfying for all scalars
{ai }∞
i=1 and all σ ⊂ N
X
X
ai ψi a i ϕi + D i∈σ c
i∈σ
X
X
a
ϕ
+
a
ψ
≤
i i
i i .
i∈σ c
i∈σ
(iv) There is a constant 0 < E satisfying
scalars
P for all {ai }∞
and
all
σ
⊂
N
so
that
if
a
ϕ
= 1
i=1
i∈σ i i
then
X
X
E≤
ai ϕi +
a i ψi .
i∈σ c
i∈σ
Proof. The implications (iii) ⇒ (ii), and (ii) ⇒ (iv) are clear.
(ii) ⇒ (iii): Given the assumptions in (ii) we compute:
X
X
X
X
a i ψi ≤ ai ϕi +
ai ψi + a i ϕi i∈σ c
i∈σ c
i∈σ
i∈σ
X
X
≤
ai ϕi +
ai ψi i∈σ
c
i∈σ
X
X
1
ai ϕi +
ai ψi + .
C i∈σ
i∈σ c
So,
X
X
X
C ai ψi ≤ ai ϕi +
ai ψi .
C + 1 i∈σc
i∈σ
i∈σ c
|ai |2 +
i∈σ
i∈σ c
i∈σ
i∈σ
≤ B2
X
X
|ai |2
i∈σ c
X
2
X
B2 a i ϕi +
ai ψi .
2
A i∈σ
i∈σ c
≤
Finally we prove (iii) ⇒ (i): For every sequence of scalars
{ai }∞
i=1 and every σ ⊂ N we have:
∞
X
i=1
|ai |2 =
X
|ai |2 +
X
|ai |2
i∈σ c
i∈σ
X
X
1 1 ai ϕi +
ai ψi ≤
A1 i∈σ
A2 i∈σc
!
X
X
1 1
ai ψi ≤ max
,
ai ϕi +
A1 A2
i∈σ c
i∈σ
X
X
1
1 1 ≤
max
,
a
ϕ
+
ai ψi i
i
D
A1 A2
i∈σ
i∈σ c
proving the lower bound. The upper bound is obvious. This
completes the proof of the theorem.
R EFERENCES
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