Unconditional Convergence Constants of Hilbert
Space Frame Expansions
Peter G. Casazza
Richard G. Lynch
Janet Tremain
Director: The Frame Research Center
Department of Mathematics
University of Missouri
Columbia, MO 65211
Email: http://www.framerc.org
Department of Mathematics
University of Missouri
Columbia, MO 65211
Email: rilynch37@gmail.com
Department of Mathematics
University of Missouri
Columbia, MO 65211
Email: tremainjc@missouri.edu
Abstract—We will prove some new, fundamental results in
frame theory by computing the unconditional constant (for all
definitions of unconditional) for the frame
p expansion of a vector
in a Hilbert space and see that it is
B/A, where A, B are
the frame bounds of the frame. It follows that tight frames
have unconditional constant one. We then generalize this to a
classification of such frames by showing that for Bessel sequences
whose frame operator can be diagonalized, the frame expansions
have unconditional constant one if and only if the Bessel sequence
is an orthogonal sum of tight frames. We then prove similar
results for cross frame expansions but here the results are no
longer a classification. We also give examples to show that our
results are best possible. These results should have been done 20
years ago but somehow we overlooked this topic.
I. I NTRODUCTION
Hilbert space frames have traditionally been used in signal
processing. But over the last few years, this has become one
of the most applied subjects in mathematics. Fundamental to
the notion of a frame is that it is a redundant sequence of
vectors Φ = {φi }i∈I in a Hilbert space H for which the frame
expansions of a vector x,
X
S(x) =
hx, φi iφi ,
i∈I
are unconditionally convergent series and the frame operator S
is a positive, self-adjoint and invertible operator on H. Until
now, no work has been done on understanding the precise
unconditional behavior of the frame expansions of vectors in
the Hilbert space. We start by reviewing some fundamental
results in this direction from [1] which show the unconditional
constants (for all standard forms of unconditional
convergence)
p
for frame expansions are of the form B/A where A, B are
the frame bounds of the frame. This means that tight frames
have 1-unconditionally convergent series for their frame expansions. We will then expand this to a classification of
Bessel sequences (whose frame operators are diagonalizable)
by showing that the frame expansions are 1-unconditional if
and only if the Bessel sequence is an orthogonal sum of tight
frames. This is surprising at first since we have not assumed
the family has any lower frame bound but conclude that locally
it does have lower frame bounds. It follows that this Bessel
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
sequence is a frame if and only if the tight frame bounds of
the orthogonal parts are uniformly bounded away from zero.
Next we will look at new results in this same direction for
cross frame expansions. That is, where {φi }i∈I and {ψi }i∈I
are Bessel sequences for which there is a constant 0 < A
satisfying for all φ ∈ H:
X
Akφk2 ≤ k
hφ, ψi iφk2 .
i∈I
We will be able to identify an upper bound for the unconditional constants for these expansions, but will see that this
does not classify these families. We will also give examples
to show that all our results are the best possible.
II. F RAME T HEORY P RELIMINARIES
We start with a brief introduction to frame theory. For a
thorough approach to the basics of frame theory, see [2],
[3]. Throughout the paper H will denote a finite or infinite
dimensional Hilbert space while HM is an M −dimensional
Hilbert space and I can represent a finite or countably infinite
index set.
Definition 2.1: A family of vectors Φ = {φi }i∈I in a Hilbert
space 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 the lower frame bound and upper frame
bound, respectively. If only B is assumed, then it is called a BBessel sequence. If A = B, it is said to be a B-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 B-Bessel sequence for H, respectively,
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 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 the positive, self-adjoint,
invertible operator defined by S := T ∗ T and satisfies
X
Sx = T ∗ T x =
hx, φi iφi
i∈I
for any x ∈ H. This is called the frame expansion of the
vector x. Also, for any x ∈ H,
X
hAx, xi ≤ hSx, xi = kT xk2 =
|hx, φi i|2 ≤ hBx, xi,
i∈I
and hence the operator inequality A · Id ≤ S ≤ B · I holds,
where ”Id” is the Identity operator. It follows that kT xk2 ≤
Bkxk2 , for all x ∈ H. Furthermore, the norm of S is kSk =
kT ∗ T k = kT k2 . Finally, note that these three operators are
also well-defined when the sequence is assumed to only be a
B-Bessel sequence.
If Φ = {φi }N
i=1 is a finite frame in an M -dimensional space
M
H , with index set I = {1, . . . , N }, then S has eigenvalues
{λj }M
j=1 and
N
M
X
X
2
kφi k =
λj .
i=1
(c) kSσ xk2 ≤ BhSx, xi ≤ B 2 kxk2 ,
(d) kSσ k ≤ kSk ≤ B,
(e) If in addition Φ is a frame with lower frame bound A,
then
X
hx, φi iφi = kSσ xk
i∈σ
r
B
≤
kSxk
rA
X
B
=
hx, φi iφi .
A
i∈I
Proof: The definition of Tσ and T gives
X
X
kTσ xk2 =
|hx, φi i|2 ≤
|hx, φi i|2 = kT xk2
i∈σ
for any x ∈ H which further gives kTσ k ≤ kT k proving
(a). Obviously, (b) follows immediately by properties of dual
operators.
To prove (c), we have first by (a) and (b) that
j=1
Furthermore, the largest and smallest eigenvalues of the frame
operator S coincIde with the optimal upper frame bound and
the optimal lower frame bound, respectively. Given a frame
Φ = {φi }i∈I for H with frame operator S, for any x ∈ H we
compute:
!
X
X
−1/2
−1/2
−1/2
−1/2
hx, S
φi iS
φi = S
hS
x, φi iφi
i∈I
i∈I
kSσ xk2
=
kTσ∗ Tσ xk2
≤
kTσ∗ k2 kTσ xk2
≤
kT ∗ k2 kT xk2
≤
BhSx, xi.
and furthermore by Cauchy-Schwarz,
BhSx, xi ≤ BkSxkkxk ≤ B 2 kxk2 ,
i∈I
=
S −1/2 S(S −1/2 x)) = x.
concluding the proof of (c) by combining inequalities.
Statement (d) is proven using
So {S −1/2 φi }i∈I is a Parseval frame.
III. H ILBERT S PACE F RAME E XPANSIONS
This section is devoted to some beginning results about what
happens when the index sets are restricted in the definition of
the analysis, synthesis, and frame operator of a sequence of
vectors.
Definition 3.1: Let {φi }i∈I be B-Bessel sequence of vectors
with analysis operator T and frame operator S. For any σ ⊂ I
denote by Tσ , Tσ∗ , and Sσ the operators
Tσ x
Tσ∗ {ci }i∈σ
= {hx, φi i}i∈σ , x ∈ H
X
=
ci φi , {ci }i∈σ ∈ `2 (σ)
i∈σ
Sσ x
X
=
hx, φi iφi ,
x∈H
i∈σ
respectively.
Remark 3.2: It follows that S = Sσ + Sσc for any choice of
σ ⊂ I.
Theorem 3.3: Let Φ = {φi }i∈I be a B-Bessel sequence for H
with analysis operator T and frame operator S, and let σ ⊂ I
and x ∈ H. Then the following holds:
√
(a) kTσ xk ≤ kT xk and
hence kTσ k ≤ kT k ≤ B,
√
(b) kTσ∗ k ≤ kT ∗ k ≤ B,
kSσ k = kTσ∗ kkTσ k ≤ kT k2 = kSk
again by (a) and (b). Since kSk = kT k2 ≤ B the bound on
the norms is also obtained.
To prove (e) recall that A·Id≤ S, and combining this with
(III) gives
kSσ xk2
≤
BhSx, xi
BhS 1/2 x, S 1/2 xi
1
1/2
1/2
≤ B
SS x, S x
A
B
=
kSxk2
A
proving the inequality by taking square roots.
In the case that the frame is tight, B = A and therefore Theorem 3.3 (5) gives the following unconditional convergence
result.
Corollary 3.4: If Φ = {φi }i∈I is a tight frame for H with
frame operator S, then for any σ ⊂ I and x ∈ H we have
X
X
hx, φi iφi = kSσ xk ≤ kSxk = hx, φi iφi .
=
i∈σ
i∈I
This can be generalized slightly to the following.
Corollary 3.5: If Φ = {φi }i∈I is a frame for H with frame
operator S, for any σ ⊂ I and x ∈ H we have
kS −1/2 Sσ xk2 ≤ kS −1/2 Sxk = kS 1/2 xk.
Proof: Choose y ∈ H so that x = S −1/2 y. Since
{S −1/2 φi }i∈I is a Parseval frame we have
X
kS −1/2 Sσ xk2 = kS −1/2
hx, φi iφi k2
i∈σ
= k
X
hS
−1/2
y, φi iS −1/2 φi k2
i∈σ
But Sσ ≤ S does not imply that Sσ2 ≤ S 2 in general.
To guarantee this, Sσ and S would need to have the same
eigenvectors, which certainly does not generally hold. What
is true, follows from Theorem 3.3 (c).
Proposition 3.6: If Φ = {φi }i∈I is a frame with frame
operator S, then for any σ ⊂ I,
Sσ2 ≤ kSkS.
Proposition 3.7: Let Φ = {φi }i∈I be a frame for H with
frame operator S. The following are equivalent:
1) There is a constant 0 < D so that for all σ ⊂ I,
X
= k
hy, S −1/2 φi iS −1/2 φi k2
Sσ2 ≤ DS 2 .
i∈σ
≤
k
X
hy, S −1/2 φi iS −1/2 φi k2
i∈I
2
= kyk = kS 1/2 xk2 .
as sought.
There are a few questions that immediately come to light
from Theorem 3.3 (5). First, one might wonder if kTσ∗ ck ≤
kT ∗ ck for all c ∈ `2 (I), similar to the inequality obtained for
Tσ . Here, it is understood that Tσ∗ c is computed by taking only
the coordinates of c with i ∈ σ. However, this is not true in
general as was shown in [1].
It was also shown in [1] that there are frames Φ = {φi }i∈I ,
with lower and upper frame bounds A and B, respectively,
and frame operator S, so that
B
kSσ xk2 ≈ kSxk2
A
holds for some σ ⊂ I, with σ 6= I, and x ∈ H, where B/A is
arbitrarily large. Hence, kSσ xk can be as large as one would
like when compared to kSxk. This warrants a discussion since
it seems like a contradiction at a first glance.
For any σ ⊂ I and x ∈ H,
X
X
hSσ x, xi =
|hx, φi i|2 ≤
|hx, φi i|2 = hSx, xi
i∈σ
i∈I
so that Sσ ≤ S. At first, it looks like this should imply the
inequality kSσ xk ≤ kSxk. However, this is not true. What
this yields is
kSσ1/2 xk2
= hSσ1/2 x, Sσ1/2 xi
= hSσ x, xi
≤
hSx, xi
= hS 1/2 x, S 1/2 xi
= kS 1/2 xk2 .
To conclude that kSσ xk ≤ kSxk, we would need Sσ2 ≤ S 2 so
that
kSσ xk2
= hSσ x, Sσ xi
= hSσ2 x, xi
≤
hS 2 x, xi
= hSx, Sxi
= kSxk2 .
2) For every x ∈ H,
X
hx, φi iφi =
kSσ xk2
i∈σ
≤
=
DkSxk2
X
hx,
φ
iφ
D
i i .
i∈I
Proof: Note that
Sσ2
2
≤ DS holds if and only if
kSσ xk2
= hSσ x, Sσ xi
= hSσ2 x, xi
≤ DhS 2 x, xi
= DhSx, Sxi
= DkSxk2 ,
giving the desired equivalence.
Remark 3.8: Proposition 3.7 combined with Theorem 3.3 (5)
gives
B
Sσ2 ≤ S 2
A
must hold for a frame with upper and lower frame bounds A
and B, respectively. In particular, Sσ2 ≤ S 2 when the frame is
tight.
We will now generalize Theorem 3.3 (5) and Corollary 3.4,
while showing that kSσ xk ≤ kSxk may hold for all σ ⊂ I
and x ∈ H, but the sequence is not a tight frame.
Proposition 3.9: Let I be a finite or infinite index set and
for each i ∈ I, let Φi = {φij }j∈Ji be a frame with lower
and upper frame bounds Ai and Bi and frame operator Si
for a finite or infinite dimensional Hilbert space Hi . Define a
Hilbert space H by
!
X
H :=
⊕Hi
i∈I
`2
and consIder the family Φ = {Φi }i∈I = {φij }i∈I,j∈Ji as
a sequence of vectors in H with frame operator S. That is,
consIder φij as a coordinate vector having zeros everywhere
but the ith component where it takes the value φij . Then for
any σ ⊂ {(i, j) : i ∈ I, j ∈ Ji } and any x ∈ H, it follows
that
r Bi
kSxk.
kSσ xk ≤ sup
Ai
i∈I
In particular, if there is a constant C > 0 so that
r
Bi
=C
Ai
for all i ∈ I, then
i∈σ
kSσ xk ≤ CkSxk.
kSσ xk ≤ kSxk
holds even though the sequence Φ as a whole may not be a
tight frame, or even a frame if inf i∈I Ai = 0.
In [1], the converse of the above theorem is shown to hold
but with the added assumption that the frame operator of the
frame is diagonalizable. This, of course, always holds in the
finite dimensional case. In the infinite dimensional case, there
are frames whose frame operators have no eigenvectors.
Theorem 3.10:[1] If Φ = {φi }i∈I is a Bessel sequence of
vectors in H with diagonalizable frame operator, then the
following are equivalent:
1) For every σ ⊂ I and for every x ∈ H,
X
X
hx,
φ
iφ
≤
hx,
φ
iφ
i i
i i
i∈I
IV. C ROSS F RAME E XPANSIONS
In this section we will look at the unconditional constants
for cross expansions of the form
X
hx, ψi iφi ,
i∈I
where {ψi }i∈I and {φi }i∈I are Bessel sequences in a Hilbert
space H and x ∈ H.
We start with the main result.
Theorem 4.1: If Φ = {φi }i∈I and Ψ = {ψi }i∈I are
Bessel sequences from H with Bessel bounds B1 and B2 ,
respectively, and there is an A > 0 so that for all x ∈ H
X
2
2
hx, ψi iφi Akxk ≤ ,
i∈I
then for any σ ⊂ I we have
2
2
X
B1 B2 P
hx, ψi iφi ≤ A i∈I hx, ψi iφi .
i∈σ
2) For every sequence of real numbers {ai }i∈I with |ai | ≤
1 for all i and for every x ∈ H,
X
X
hx,
φ
iφ
a
hx,
φ
iφ
≤
i i
i
i i
Proof: We compute
X
2
hx,
ψ
iφ
i i
3) For any sequence E = {i }i∈I with i ∈ {−1, 1} and
for every x ∈ H,
X
X
hx,
φ
iφ
≤
hx,
φ
iφ
i
i i
i i
4) There is a partition {µj }j∈J of I satisfying:
a) For every j ∈ J, {φi }i∈µj is a tight frame,
b) For any j1 , j2 ∈ J with j1 6= j2 , it follows that
hφk1 , φk2 i = 0 for any k1 ∈ µj1 and k2 ∈ µj2 .
In other words,


X
H=
⊕span{φi : i ∈ µj }
j∈J
`2
where {φi }i∈µj is a tight frame for span{φi : i ∈ µj }.
Hence, Φ is a frame if and only if the infimum of the tight
frame bounds is not equal to zero.
We have seen that tight frames have unconditional constant
1 with respect to their frame expansions. Based on these
results, we could wonder if tight frames are 1-unconditional
with respect to expansions of any family of coefficients. As
shown in [1], this occurs only in the case where the frame
vectors are orthogonal.
X
≤ B1
X
|hx, ψi i|2
i∈σ
|hx, ψi i|2
i∈I
≤ B1 B2 kxk2
X
2
B1 B2 ≤
hx, ψi iφi .
A
i∈I
i∈I
i∈I
≤ B1
i∈σ
i∈I
i∈I
i∈I
(2) Φ is an orthogonal basis.
Furthermore, if C = 1, then each frame Φi is tight with
possibly different tightness factors Ai , and
i∈σ
Proposition 3.11: Let Φ = {φi }i∈I be a tight frame for H.
The following are equivalent:
(1) For any σ ⊂ I and scalars {ai }i∈I we have:
X
X
ai φi .
ai φi ≤ Corollary 4.2: If Φ = {φi }i∈I is a Parseval frame and Ψ =
{ψi }i∈I is a dual frame to Φ with upper frame bound B, then
for any σ ⊂ I we have
X
2
2
P
hx,
ψ
iφ
≤
B
hx,
ψ
iφ
i i
i i .
i∈I
i∈σ
Proof: Our assumptions imply in Theorem 4.1: A = B1 = 1,
and B = B2 .
Remark 3.3: The inequality above actually implies that Φ is a
frame and Ψ is a frame sequence (I.e. a frame for a subspace).
If T, U : `2 → H are the synthesis operators of Φ and Ψ,
respectively, then this condition says that T U ∗ is invertible
and each x ∈ H has representation
X
x = (T U ∗ )−1 (T U ∗ )x =
hx, ψi i(T U ∗ )−1 φi .
i∈I
From here, we get that {ψi }i∈I is a frame with dual
frame {(T U ∗ )−1 φi }i∈I by Lemma 5.6.2 of [3]. Furthermore,
{φi }i∈I is a frame sequence by a symmetric argument.
A. Examples
We will now give a series of examples showing the limitations of the theorem. First, we will see that the lower bound
A in the cross frame expansion is necessary to draw any real
conslusions.
this result is true for frames for which the frame operator is diagonalizable. We then extended this to cross frame expansions
where for all x ∈ H,
X
Akxk2 ≤ hx, ψi iφi ,
i∈I
Example 4.4 Let {ei } be an orthonormal basis. Let
ψi =
1
ei , and φi = ei .
i
The sums here are 1-unconditional but we do not have the
lower bound A in Theoerem 3.1. One can actually let
ψi =
a
ei
i
and
b
φi = e i
i
where a, b > 0 are arbitrary. This gives two Bessel sequences
of arbitrary bounds having mixed unconditional constant one.
This just says that this theorem does not classify the Bessel
sequences satisfying it. We can ask if the constant B1 B2 /A
is “close” to being the best for at least some cases where
condition (1) holds. The answer to this is yes:
Example 4.5: Let F = {fi }N
i=1 be a Parseval frame and
let Φ = {φi }i∈I = {F, F, . . . , F }, where F is repeated
K + 1 times and K is a positive even integer. Notice that
Φ is a (K + 1)-tight frame with dual Ψ = {ψi }i∈I =
{F, F, −F, . . . , F, −F }, where F appears (K + 1) times in
Ψ, but has alternating signs for the last K. Notice that Ψ is
also a (K + 1)-tight frame. Thus, B1 = B2 = (K + 1) and
furthermore, A = 1 as given in (1) since the two frames form
a dual pair. I.e. For any x ∈ H we have
X
hx, ψi iφi = x.
i∈I
Letting σ be the indices where F is positive gives that
X
2
2
K
hx, ψi iφi =
+ 1 kxk2
2
i∈σ
≈
=
(K + 1)2 kxk2
X
2
B1 B2 hx, ψi iφi .
A
i∈I
V. C ONCLUSION
We have seen that Hilbert
space frame expansions have
p
B/A where A, B are the frame
unconditional constant
bounds of the frame. This then generalizes to an orthogonal
sum of Hilbert space frames with frame bounds Aj , Bj for
j ∈ J yielding unconditional constant
r
Bi
sup
.
Ai
i∈I
So a frame which is an orthogonal sum of tight frames has
unconditional constant one. We then saw that the converse of
where {φi }i∈I and {ψi }i∈I are Bessel sequences with Bessel
bounds B1 , B2 respectively. In this case the unconditional
constant is:
B1 B2
.
A
Also in this case, we see that this is not a classification of
dual frames, but in general these constants are necessary.
Finally, we gave examples showing that all these results are
best possible.
ACKNOWLEDGMENT
The authors are indebted to meticulous refereeing which
greatly improved the paper.
The authors were supported by:
NSF DMS 1307685; and NSF ATD 1042701 and 1321779;
and AFOSR FA9550-11-1-0245.
R EFERENCES
[1] T. Bemrose, P.G. Casazza, R.G. Lynch, The unconditional constant for
Hilbert space frame expansions, Preprint.
[2] P.G. Casazza and G. Kutyniok, Finite frames: Theory and applications,
Springer (2013).
[3] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhaüser,
Boston (2003).