Nullspaces and frames

Abstract

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.

Full text

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arXiv:math/0411433v1 [math.FA] 19 Nov 2004
Nullspaces and frames∗
J. Antezana , G. Corach , M. Ruiz and D. Stojanoff
February 1, 2008
Jorge Antezana, Mariano Ruiz and Demetrio Stojanoff
Depto. de Matem´ atica, FCE-UNLP, La Plata, Argentina and IAM-CONICET
e-mail: antezana@mate.unlp.edu.ar, mruiz@mate.unlp.edu.ar
and demetrio@mate.unlp.edu.ar
Gustavo Corach (corresponding author)
Depto. de Matem´ atica, FI-UBA and IAM-CONICET,
Saavedra 15, Piso 3 (1083),
Ciudad Aut´ onoma de Buenos Aires, Argentina.
Phone: (54) (11) 4954 - 6781
Fax: (54) (11) 4954 - 6782
e-mail: gcorach@fi.uba.ar
Keywords: frames, generalized inverses, Riesz frames, angles.
2000 AMS Subject Classifications: Primary 42C15, 47A05.
Abstract
In this paper we give new characterizations of Riesz and conditional Riesz frames
in terms of the properties of the nullspace of their synthesis operators. On the other
hand, we also study the oblique dual frames whose coefficients in the reconstruction
formula minimize different weighted norms.
1Introduction
Frames were introduced by Duffin and Schaeffer [16] in the context of nonharmonic Fourier
series, and they have been intensively applied in wavelet and frequence analysis theories
since the work of Daubechies, Grossmann and Meyer [14]. Today, frame-like expansions are
fundamental in a wide range of disciplines (see for example [16], [17] or [25]), including the
analysis and design of oversampled filter banks and error corrections codes.
A frame is a redundant set of vectors in a Hilbert space that leads to expansions of vectors
(signals) in terms of the frame elements. More precisely, a sequence of vectors F = {fn}n∈N
∗Partially supported by CONICET (PIP 2083/00), UBACYT I030, UNLP (11 X350) and ANPCYT
(PICT03-9521)
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in a (separable) Hilbert space H is a frame (for H) if there exist numbers A,B > 0 such
that, for every f ∈ H,
A?f?2≤
|?f,fn?|2≤ B?f?2.
?
n∈N
(1)
Associated with each frame there exists an operator T : ℓ2→ H defined by T(en) = fn,
where B = {en}n∈Ndenotes the canonical basis of ℓ2; T is called the synthesis operator of
F.
The results of this paper can be divided in two parts. The main results of the first part
are devoted to the study of Riesz frames and conditional Riesz frames through the structure
and geometric properties of the nullspace of their synthesis operators. Riesz and conditional
Riesz frames were introduced by Christensen in [9] (see definitions in section 3). These frames
are important because they behave well with respect to the projection method. In general,
frame theory describes how to choose the corresponding coefficients to expand a given vector
in terms of the frame vectors. However, in applications, to obtain these coefficient requires
the inversion of an operator on H. The projection method was introduced by Christensen
in [7] to avoid this problem. We refer the interested reader to [6], [7], [9] or [10] for more
information about the projection method. In [1] we found a characterization of Riesz frames
by studying the nullspace of the synthesis operator. Namely, if the nullspace N(T) has a
certain geometric property of compatibility with the closed subspaces spanned by subsets
of B, then F is a Riesz frame, and conversely. In section 3, we extend these results for
conditional Riesz frames and give some new characterizations in terms of angles.
Throughout the second part of this work we study the so-called oblique dual frames. Let
{fn}n∈N be a frame for the closed subspace W ⊆ H, and let M ⊆ H be another closed
subspace such that H = W ˙ +M⊥(˙ + means a non necessarily orthogonal direct sum). The
sequence {gn}n∈Nin M is an oblique dual frame of {fn}n∈N(see Li [21] or Li and Ogawa
[22] and [23]) if
∞
?
Among the oblique dual frames, there exists a particular class with the minimal norm prop-
erty. Recall that a dual frame {gn}n∈Nhas the minimal norm property if the coefficients
{?f, gn?}n∈Nthat appear in the reconstruction formula have minimal ℓ2norm.
If B = {en}n∈Ndenotes the canonical orthonormal basis for ℓ2and T is the synthesis
operator of {fn}n∈N, then Christensen and Eldar [11] proved that the minimal norm oblique
dual frames have the form
{gn}n∈N=?B(T∗B)†en
where B is any bounded operator with R(B) = M. From the point of view of sampling
theory, the operator B can be interpreted as the synthesis operator associated to the frame
used to sample the signals.
In this work, we are interested in duals frames which lead to reconstruction coefficients
that have minimal norm, but with respect to some weighted norms. Recall that weighted
norms in ℓ2arise from inner products obtained by perturbing the original one with invertible
positive operators which are diagonal in the canonical basis. In section 4 we give explicit
formulae for dual frames which minimize a given weighted norm, and we prove that in the
f =
n=1
?f, gn?fn
∀ f ∈ W.
?
n∈N,
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case of Riesz frames, if the sampling frame is fixed, then the norms of the preframes operators
corresponding to the dual frames which minimize the different weighted norms are uniformly
bounded from above.
We thank Ole Christensen for his useful comments.
2Preliminaries
Let H be a separable Hilbert space and L(H) the algebra of bounded linear operators on
H. Gl(H) denotes the group of invertible operators in L(H), and Gl(H)+the set of positive
definite invertible operators on H. For an operator A ∈ L(H), R(A) denotes the range
of A, N(A) the nullspace of A, σ(A) the spectrum of A, A∗the adjoint of A, ρ(A) the
spectral radius of A and ?A? the operator norm of A; if R(A) is closed, A†is the Moore-
Penrose pseudoinverse of A. We use the fact that A is an isometry (resp. coisometry) if
A∗A = I (resp. AA∗= I). Given a closed subspace M of H, PMdenotes the orthogonal
(i.e., selfadjoint) projection onto M. If B ∈ L(H) satisfies PMBPM= B, we consider the
compression of B to M, (i.e., the restriction of B to M, which is an operator on M), and
we say that we consider B as acting on M. Given a subspace M of H, its unit ball is
denoted by M1, and its closure by M or cl(M). If N is another subspace of H, we denote
M ⊖ N := M ∩ N⊥. If M ∩ N = {0}, we denote by M˙ +N the (direct) sum of the two
subspaces. If the sum is orthogonal, we write M ⊕ N. The distance between two subsets
M and N of H is d(M, N) = inf{?x − y? : x ∈ M y ∈ N}.
2.1Angle between closed subspaces
We shall recall the definition of angle between closed subspaces of H. We refer the reader to
the nice survey of Deutsch [15] and the books by Kato [19] and Havin and J¨ oricke [18] for
details and proofs.
Definition 2.1. Given two closed subspaces M and N, let˜ N = N ⊖ (M ∩ N) and ˜
M ⊖ (M ∩ N). The angle between M and N is the angle in [0,π/2] whose cosine is
M =
c[M, N ] = sup{|?x, y?| : x ∈˜
M, y ∈˜ N and ?x? = ?y? = 1}
The sine of this angle is denoted by s[M, N ].
Now, we state some known results concerning angles and closed range operators (see [15]).
Proposition 2.2. Let M and N be two closed subspaces of H. Then
?
2. c[M, N ] < 1 if and only if M + N is closed.
3. c[M, N ] = c?M⊥, N⊥?
4. c[M, N ] = ?PMP˜ N? = ?P ˜
1. c[M, N ] = c[N, M] = c
˜
M, N
?
= c
?
M,˜ N
?
.
MPN? = ?PMPNP(M∩N)⊥? = ?PMPN− PM∩N?
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Proposition 2.3 (Bouldin [2]; see also [15]). Let A,B ∈ L(H) such that R(A) and R(B)
are closed. Then, AB has closed range if and only if c[R(B), N(A)] < 1.
Proposition 2.4 (Kayalar-Weinert [20]; see also [15]). Let P and Q two orthogonal
projections defined on H. Then,
?(PQ)k− P ∧ Q? = c[R(P), R(Q)]2k−1
where P ∧ Q is the orthogonal projection onto R(P) ∩ R(Q).
Finally, we give a characterization of s[M, N ] in terms of distances:
Proposition 2.5. Let M and N be to closed subspaces of H. Denote˜ N = N ⊖ (M ∩ N)
and ˜
M = M ⊖ (M ∩ N). Then
?
Proof. By Proposition 2.2, we can suppose that M ∩ N = {0}, i.e., M =
definition of the sine and Proposition 2.2, s[M, N ]2= 1 − ?PMPN?2. On the other hand,
as d(x, N) = ?PN⊥ x? for every x ∈ H, we have that
s[M, N ] = d
˜
M1, N
?
= d
?
˜ N1, M
?
.
˜
M. By the
d(M1, N)2
= inf{?PN⊥ x?2: x ∈ M1} = inf{1 − ?PNx?2: x ∈ M1}
= 1 − sup{?PNx?2: x ∈ M1} = 1 − ?PNPM?2= 1 − ?PMPN?2.
?
2.2The reduced minimum modulus
Definition 2.6. The reduced minimum modulus γ(T) of an operator T ∈ L(H) is defined
by
γ(T) = inf{?Tx? : ?x? = 1 , x ∈ N(T)⊥}
(2)
It is well known that γ(T) = γ(T∗) = γ(T∗T)1/2. Also, it can be shown that an operator T
has closed range if and only if γ(T) > 0. In this case, γ(T) = ?T†?−1.
The following result is an easy consequence of equation (2):
Lemma 2.7. Let B ∈ L(H) with B invertible. Then,
?B−1?−1γ(T) ≤ γ(BT) ≤ ?B?γ(T).
Moreover, the same formula follows, replacing ?B−1?−1by γ(B), if R(B) is closed and
R(T) ⊆ N(B)⊥.
Lemma 2.8. Let T ∈ L(H) be a partial isometry (i.e., TT∗is a projection), M a closed
subspace of H and PMthe orthogonal projection onto M. Then
γ(TPM) = s[N(T), M] .
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Proof. Denote N = N(T) and R = N⊥. Since T acts isometrically on R, it is clear by
equation (2) that
γ(TPM) = γ(TPRPM) = γ(PRPM).
Since N(PRPM) = M⊥⊕ (M ∩ N), it follows that N(PRPM)⊥= M ∩ (M ∩ N)⊥= ˜
Then, by Proposition 2.5,
M.
γ(PRPM) = inf
x∈˜
M1?PRx? = inf
x∈˜
M1d(x, N) = d
?
˜
M1, N
?
= s[N, M] .
?
The next result was proved in [1]. We include a short proof for the sake of completeness.
Proposition 2.9. If T ∈ L(H) has closed range and M is a closed subspace of H such that
c[N(T), M] < 1 (so that TPMhas closed range), then
γ(T) s[N(T), M] ≤ γ(TPM) ≤ ?T? s[N(T), M]. (3)
Proof. Take B = |T∗| = (TT∗)1/2. It is well known that R(B) = R(T) which is closed by
hypothesis. It is easy to see that γ(T) = γ(B) and ?B? = ?T?. Also, B†T is a coisometry,
with the same nullspace as T. So, by Lemma 2.8, γ(B†TPM) = s[N(T), M]. Now, using
Lemma 2.7 for B and B†TPMand the fact that BB†TPM= PR(T)TPM= TPM, we get
γ(T) s[N(T), M] ≤ γ(TPM) ≤ ?T? s[N(T), M],
because R(B) = R(B†), so that R(B†TPM) ⊆ R(B) = N(B)⊥.
?
Remark 2.10. With the same ideas, the following formulae generalizing Lemma 2.8 and
Proposition 2.9, can be proved.
1. Let U,V ∈ L(H) be partial isometries. Then, γ(UV ) = s[N(U), R(V )].
2. If A,B ∈ L(H) have closed ranges, then
γ(A)γ(B) s[N(A), R(B)] ≤ γ(AB) ≤ ?A??B? s[N(A), R(B)].
Note that the first inequality implies Proposition 2.3.
In particular, this gives the following formula for the sine of an angle: given M and N two
closed subspaces of H, it holds
s[N, M] = γ(PN⊥PM).
△
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2.3Frames
We introduce some basic facts about frames in Hilbert spaces. For complete descriptions of
frame theory and applications, the reader is referred to the survey by Heil and Walnut [17]
or the books by Young [25] and Christensen [10].
Definition 2.11. Let H be a separable Hilbert space, and F = {fn}n∈Na sequence in H.
1. F is called a frame if there exist numbers A,B > 0 such that, for every f ∈ H,
?
2. The optimal constants A,B for equation (4) are called the frame bounds for F.
A?f?2≤
n∈N
|?f,fn?|2≤ B?f?2
(4)
3. The frame F is called tight if A = B, and Parseval if A = B = 1.
4. Associated with F there exist an operator T : ℓ2→ H such that T(en) = fnwhere
{en}n∈Ndenotes the canonical basis of ℓ2. This operator is called the synthesis operator
of F. For finite frames we assume that the domain of the synthesis operator is Cm
where m is the number of vectors of the frame.
Remark 2.12. Let F = {fn}n∈Nbe a frame in H and T its synthesis operator.
1. The frame bounds of F can be computed in terms of the synthesis operator
A = γ(T)2
andB = ?T?2.(5)
2. The adjoint T∗∈ L(H,ℓ2) of T, is given by T∗(x) =
?
n∈N
?x,fn?en, x ∈ H. It is called
the analysis operator for F.
3. The operator S = TT∗is usually called frame operator and it is easy to see that
?
It follows from (4) that A.I ≤ S ≤ B.I, so that S ∈ Gl(H)+. Moreover, the optimal
constants A,B for equation (4) are
Sf =
n∈N
?f,fn?fn
f ∈ H.(6)
B = ?S? = ρ(S) andA = γ(S) = ?S−1?−1= min{λ : λ ∈ σ(S)}.
Finally, from (6) we get
f =
?
n∈N
?f,S−1fn
?fn
∀ f ∈ H.
4. The numbers {?f,S−1fn
?} are called the frame coefficients of f. They have the fol-
?
The frame {S−1fn}n∈Nis called canonical dual frame. We shall return to dual frames
in section 4.
lowing optimal property: if f =?
n∈Ncnfn, for a sequence (cn)n∈N, then
|?f,S−1fn
n∈N
?|2≤
?
n∈N
|cn|2.
△
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3Riesz frames and conditional Riesz frames.
It was remarked by Christensen [10] , p. 65, that given a frame F = {fn}n∈N, in practice
it can be difficult to use the frame decomposition f =??f,S−1fn
of the advantages of Riesz bases, Christensen introduced in [7] the projection method,
approximating S and S−1by finite rank operators, acting on certain finite dimensional spaces
Hnapproaching H. Later on, Christensen [9] introduced two special classes of frames, namely
Riesz frames and conditional Riesz frames, which are well adapted to some of these
problems (see also [3], [4], and [5]).
?fnbecause it requires
the calculation of S−1or, at least, the frame coefficients
?f,S−1fn
?. In order to get some
We need to fix some notations: Let B = {en}n∈Nbe the canonical orthonormal basis of ℓ2
and I ⊆ N.
1. We denote MI = span{en: n ∈ I} and PI = PMI, the orthogonal projection onto
MI.
2. If I = In:= {1,2,...,n}, we put Mnfor MI.
3. Given N a closed subspace of ℓ2, we denote Nn= N ∩ Mn, n ∈ N.
4. If F = {fn}n∈Nis a frame for H, we denote by FI= {fn}n∈I.
5. We say that FIis a frame sequence if it is a frame for span{FI}.
6. FIis called a subframe of F if it is itself a frame for H.
Recall the definitions of Riesz frames and conditional Riesz frames.
Definition 3.1. A frame F = {fn}n∈N is called a Riesz frame if there exists A,B > 0
such that, for every I ⊂ N, the subfamily FI is a frame sequence with bounds A,B (not
necessarily optimal).
The sequence F is called a conditional Riesz frame if there are common bounds for the
frame sequences FIn, where {In}∞
for every n ∈ N andIn= N.
n=1is a sequence of finite subsets of N such that In⊆ In+1
?
n∈N
Remark 3.2. Let F be a frame, and T its synthesis operator. Given I ⊆ N, then FI is
a frame sequenceif and only if R(TPI) is closed, and FI is a subframe if and only if
R(TPI) = H. In both cases the frame bounds for FI are A = γ(TPI)2and B = ?TPI?2.
Using these facts we get an equivalent definition of Riesz frames: F is a Riesz frame if there
exists ε > 0 such that γ(TPI) ≥ ε for every I ⊆ N.
△
Proposition 2.9 can be used to characterize Riesz frames in terms of the angles between the
nullspace of the synthesis operator T and the closed subspaces of ℓ2which are spanned by
subsets of B.
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Proposition 3.3. Let F = {fn}n∈Nbe a frame, and T be its synthesis operator. Let N =
N(T). Then F is a Riesz frame if and only if
c = sup
I⊆N
c[N, MI] < 1.(7)
Proof. By Proposition 2.3, TPI has closed range iff c[N, MI] < 1. By Proposition 2.9,
γ(TPI) has an uniform lower bound if and only if there exists a constant c < 1 such that,
for every I ⊆ N, c[N, MI] ≤ c.
?
Remark 3.4. Let N be a closed subspace of ℓ2and B = {en}n∈Nbe the canonical orthonor-
mal basis of ℓ2. If equation (7) holds, following the terminology of [1], we say that N is
B-compatible.
△
In the following Proposition, we state a characterization of B- compatible subspaces of H,
proved in [1].
Proposition 3.5. Let N be a closed subspace of ℓ2and let B = {ek}k∈Nbe the canonical
orthonormal basis of ℓ2. For n ∈ N, denote by cn= sup
J⊆In
c[Nn, MJ]. Then the following
conditions are equivalent:
1. N is B-compatible.
2. c = sup
n∈N
c[N, Mn] < 1, and sup
n∈N
cn< 1.
3. cl??
4. There exists a constant c < 1 such that c[N, HI] ≤ c for every finite subset I of N
with N ∩ MI= {0}.
n∈NNn
?= N and sup
n∈N
cn< 1.
?
Proposition 3.5 can be “translated” to frame language to get a characterization of Riesz
frames, similar to the one obtained by Christensen and Lindner in [13]:
Theorem 3.6. Let F = {fn}n∈Nbe a frame and T its synthesis operator. Denote N = N(T).
Then the following conditions are equivalent:
1. F is a Riesz frame.
2. N is B-compatible.
3. There exists an uniform lower frame bound for every finite linearly independent frame
sequence FJ, J ⊂ N.
4. There exists d > 0 such that γ(TPJ) ≥ d, for every J ∈ N finite such that N ∩ MJ=
{0}.
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Proof. If I is a finite subset of N then MI∩N = {0} if and only if FIis linearly independent.
Then, conditions 3 and 4 are equivalent. By Propositions 2.9 and 3.5, they are also equivalent
to the B-compatibility of N.
Suppose that there exists a constant d such that 0 < d ≤ γ(TPMI) for every finite subset
I ⊆ N such that MI∩ N = {0}. This is equivalent to saying that there is a constant c < 1
such that c[N, MI] ≤ c for such kind of sets I. Using Propositions 3.3 and 3.5, we conclude
that F is a Riesz frame. The converse is clear.
?
Now, we consider conditional Riesz frames. First of all, we state a result for this class of
frames which is similar to Proposition 3.3, and whose proof follows essentially the same lines.
Proposition 3.7. Let F = {fn}n∈Nand N the nullspace of its synthesis operator. Then F
is a conditional Riesz frame if and only if there exists a sequence {In} of finite subsets of
N such that In⊆ In+1,
?
n∈N
In= N
andc = sup
n∈N
c[N, MIn] < 1 , n ∈ N.(8)
As a corollary of this Proposition we get the following result:
Proposition 3.8. Let F be a conditional Riesz frame, and T its synthesis operator for F.
Denote N = N(T). Then
?∞
n=1
cl
?
Nn
?
= N. (9)
In order to prove this Proposition, we need the following technical Lemma.
Lemma 3.9. Let N be a closed subspace of ℓ2, a constant c < 1 and a sequence {In} of
finite subsets of N such that In⊆ In+1,?
cl
N ∩ MIn
n∈NIn= N and c[N, MIn] ≤ c, for every n ∈ N.
?
Then
??
n∈N
= N .
Proof. Denote Qn= PIn, n ∈ N. The assertion of the Lemma is equivalent to
PN∧ Qn
SOT
ր
n→∞PN .
Let x ∈ ℓ2be a unit vector and let ε > 0. Let k ∈ N such that c2k−1≤ε
2.4, for every n ≥ 1 it holds that
???(PNQn)k− PN∧ Qn
On the other hand, since QnPN
− − − →
n→∞
on bounded sets (see, for example, 2.3.2 of [24]), there exists n0≥ 1 such that, for every
n ≥ n0,
???
9
2. By Proposition
??? ≤ε
2.
SOT
PN and the function f(x) = xkis SOT-continuous
?
(QnPN)k− PN
?
x
??? <ε
2.

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Then, for every n ≥ n0,
?(PN− PN∧ Qn) x? ≤
???
?
PN− (PNQn)k?
x
??? +
???
?
(PNQn)k− PN∧ Qn
?
x
??? < ε .
?
Proof of Proposition 3.8. Since F is a conditional Riesz frame, there exist c < 1 and a
sequence {In} of finite subsets of N such that In⊆ In+1,?
there exists m ∈ N such that In⊆ Im= {1,2,...,m}. Thus,?
As a consequence of Proposition 3.8 we obtain the following Corollaries.
n∈NIn= N and c[N, MIn] ≤ c,
for every n ∈ N. By Lemma 3.9,?
?
n∈NN ∩ MInis dense in N. Finally, for every n ∈ N,
n∈NN ∩ MIn⊆?
m∈NNm.
Corollary 3.10. Let F be a conditional Riesz frame with synthesis operator T and suppose
that dimN(T) < ∞. Then F is a Riesz frame. Moreover, there exists m ∈ N such that
N(T) ⊆ Mm.
Proof. Denote by N = N(T). By Proposition 3.8, N satisfies equation (9). Since dimN <
∞, then there exists m ∈ N such that N = N(T) ⊆ Mm. Thus, in the terminology of
Proposition 3.5, if cn = sup
J⊆In
Proposition 3.5, F is a Riesz frame.
c[Nn, MJ], then cn = cm for every n ≥ m. Therefore, by
?
Corollary 3.11. Let F = {fn}n∈Nbe a conditional Riesz frame. Given n ∈ N, denote by
Snthe frame operator of {fk}n
all frame subsequences of {S−1/2
n
fk}n
k=1and let Anbe the minimum of the lower frame bounds of
k=1. If infnAn> 0, then F is a Riesz frame.
Proof. Let T be the synthesis operator of F and N = N(T). For each n ∈ N, denote Fn=
{fk}n
can be considered, modulo an unitary operator, as the synthesis operator of Fn. In this
way, it holds that Sn = TPnT∗. Also note that {S−1/2
N(TPn) = N(S−1/2
n
TPn) = M ∩ Hn= Mn. So, by Lemma 2.8, if J ⊂ {1,...,n}, the lower
frame bound AJ of {S−1/2
n
fk}k∈J satisfies AJ = 1 − c[Mn, HJ]2. Using Propositions 3.8
and 3.5, the corollary follows.
k=1, Bn= {e1,...,en} and Pn= PMn. Note that TPn: Mn→ span{fk: k = 1,...,n}
n
fk}n
k=1is a Parseval frame, and
?
A counterexample
The nullspace N of the synthesis operator of a conditional Riesz frame has the property of
“density”: cl(?∞
such that its synthesis nullspace N satisfies cl(?∞
We shall prove the assertion in an indirect way, by using Proposition 3.7 and the following
fact: if N is a closed subspace of ℓ2such that dimN⊥= ∞, then there exists a frame F
with synthesis operator T such that N = N(T).
n=1Nn) = N, where Nnis N ∩Mn. In the following example we show that
the converse is not true, i.e., we construct a frame which is not a conditional Riesz frame
n=1Nn) = N.
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Example 3.12. Given r > 1, if B = {en}n∈Ndenotes the canonical basis of ℓ2, let us define
the following orthogonal system:
x1= e1− re2+1
re3+1
r5e7+1
r2e4+1
r6e8+1
r3e5+1
r7e9+1
r4e6
x2= e5− re6+1
r8e10
...
xn= e4n−3− re4n−2+
1
r4n−3e4n−1+
1
r4n−2e4n+
1
r4n−1e4n+1+
1
r4ne4n+2.
Let N be the closed subspace generated by {xn}n∈N. By construction, cl(?∞
exists a frame F such that the nullspace of its synthesis operator is N. We claim that this
frame is not a conditional Riesz frame. By Proposition 3.7, it suffices to verify that for every
sequence J1⊆ J2⊆ J3⊆ ... ⊆ Jnր N, it holds that c[N, MJk] − − − →
a sequence {Jk}k∈Nand take 0 < ε < 1.
Since ?xn?2≤ 1 + r2+
r8n−6for every n ∈ N, there exists n0∈ N such that
n=1Nn) = N.
Moreover {e4n−1− re4n : n ∈ N} ⊂ N⊥, so dimN⊥= ∞. By the remarks above, there
k→∞1. Hence, fix such
4
1 − ε <1 + r2
?xn?2
∀ n ≥ n0.
Note that, for y ∈ N and i ∈ N, if Mi= span{e4i−3,e4i−2}, then
?y,xi? = 0 ⇐⇒ PMiy = 0, (10)
because PMixj?= 0 if and only if j = i. Let k ∈ N be such that
j = max
?
i ∈ N : PMi(N ∩ MJk) ?= 0
?
≥ n0.
By equation (10), xh∈ (N ∩ MJk)⊥for every h > j. In particular, xj+1∈ N ⊖ (N ∩ MJk)
and
1 + r2
?xj+1?2≤?PJkxj+1?2
?xj+1?2
A similar argument shows that 1 − ε ≤ c[N, MJm], for every m ≥ k. This implies that
liminf
1 − ε <
≤
?
xj+1
?xj+1?,
PJkxj+1
?PJkxj+1?
?
≤ c[N, MJk]
n→∞c[N, MJn] ≥ 1 − ε. Finally, as ε is arbitrary, we get c[N, MJk] − − − →
k→∞1.
△
4 Weighted dual frames.
Let F = {fn}n∈Nbe a fixed frame for a closed subspace W of H and let M ⊆ H be another
closed subspace such that H = W ˙ +M⊥. As we have mentioned in the introduction, an
oblique dual frame of F in M is a frame G = {gn}n∈Nfor M such that for every f ∈ W it
holds that
f =
∞
?
n=1
?f, gn?fn
∀ f ∈ W.(11)
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Such a dual frame has the minimal norm property if for every f ∈ W the coefficients
{?f, gn?}n∈Nhave minimal ℓ2norm. Christensen and Eldar proved in [12] that the duals
frames with the minimal norm property have the form
{gn}n∈N=?B(T∗B)†en
where {en}n∈Ndenote the canonical orthonormal basis of ℓ2, and B is a bounded operator
with R(B) = M.
On the other hand, let D(ℓ2) be the set of all D ∈ Gl(ℓ2)+which are diagonal in the canonical
basis {en}n∈N. Each D ∈ D(ℓ2) defines an inner product ?·, ·?Dby means of
?
n∈N,(12)
?x, y?D= ?Dx, y? ,x,y ∈ ℓ2.
This inner product induces a weighted norm ? · ?Dwhich is equivalent to the original one.
In this section, we are interested in dual frames such that their coefficients in the reconstruc-
tion formula (11) minimize different weighted norms. We shall give explicit formulae for this
class of dual frames that we call weighted dual frames. We also consider the particular case
of weighted dual frames associated to a Riesz frame.
First of all, let us recall some preliminary facts on generalized inverse s.
Definition 4.1. Given two Hilbert spaces H and K, let A ∈ L(H,K) be an operator with
closed range. We say that B ∈ L(K,H) is a generalized inverse of A if ABA = A and
BAB = B.
Remarks 4.2. Let A ∈ L(H,K) with closed range, and let B ∈ L(K,H) be a generalized
inverse of A. Then
1. Both AB and BA are oblique projections, i.e. idempotent operators.
2. R(B) is also closed.
3. The idempotent AB and BA induce decompositions of the Hilbert spaces H and K:
H = N(A)˙ +R(B) and K = R(A)˙ +N(B).
4. If (AB)∗= AB and (BA)∗= BA, then B is called the Moore-Penrose generalized
inverse for A. It is usually denoted by A†. In this case, AA†is the orthogonal projection
onto R(A) and A†A is the orthogonal projection onto N(A)⊥.
△
Among the generalized inverses of an operator A ∈ L(ℓ2,H), the following ones will be
particularly important for us. In order to clarify the next statement, given a subspace T of
ℓ2and D ∈ D(ℓ2), the orthogonal complement of T with respect to the the inner product
?·, ·?Dwill be denoted by T⊥D.
Lemma 4.3. Let A ∈ L(ℓ2,H) be an operator with closed range, and D ∈ D(ℓ2). Then, the
operator χD(A) = D−1/2(AD−1/2)†is a generalized inverse of A such that χD(A)A is the
orthogonal projection with respect to the weighted inner product ?·, ·?Donto N(A)⊥D.
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Proof. Since R(AD1/2) = R(A) it follows that
A χD(A) A = PR(AD1/2)A = A.
On the other hand,
χD(A) A χD(A) = D−1/2(AD−1/2)†AD−1/2(AD−1/2)†= D−1/2(AD−1/2)†=χD(A).
Finally, some easy computation shows that an oblique projection Q is D-orthogonal if and
only if DQ is selfadjoint. In our case
D?χD(A)A?= D1/2(AD−1/2)†A = D1/2?
which is clearly selfadjoint. Therefore, χD(A)A is a D-orthogonal projection and clearly
N?χD(A)A?= N(A).
Now, we are ready to give the explicit form of weighted dual frames.
D−1/2A∗(ADA∗)†?
A = A∗(ADA∗)†A,
?
Proposition 4.4. Let F = {fn}n∈Nbe a fixed frame for a closed subspace W of H, T its
synthesis operator and let M be another closed subspace of H such that H = W ˙ +M⊥.
Then, given D ∈ D(ℓ2), the oblique dual frames such that for every f ∈ W their coefficient
in the reconstruction formula minimize the weighted norm ? · ?Dhave the form
G = {gn}n∈N= {B(D1/2T∗B)†D1/2en}n∈N
where {en}n∈Ndenotes the canonical orthonormal basis of ℓ2and B ∈ L(ℓ2,H) is any operator
with R(B) = M.
Proof. Fix B ∈ L(ℓ2,H) with range M and let?T = B(D1/2T∗B)†D1/2. First of all, note
In order to prove that G is an oblique dual frame it is enough to prove that T?T∗is an
on one hand
?
= T= (T?T∗),
which shows that T?T∗is a projection. On the other hand, since N(D−1/2(B∗TD−1/2)†) =
projection onto W with nullspace M⊥.
Finally, in order to prove that the reconstruction coefficients minimize the weighted norm
? · ?Dwe have to prove that R(?T∗) ⊆ N(T)⊥D. But, using the notation of Lemma 4.3, we
N(T)⊥D.
that N(D1/2T∗B) = N(B). So, R(?T) = R(B) = M and therefore G is a frame.
oblique projection onto W. Actually, T?T∗is the projection onto W parallel to M⊥. Indeed,
(T?T∗)2= TD−1/2(B∗TD−1/2)†B∗?2
= TD−1/2?
(B∗TD−1/2)†(B∗TD−1/2)(B∗TD−1/2)†?
B∗
?
D−1/2(B∗TD−1/2)†B∗?
R(B∗T)⊥= R(B)⊥and R(D−1/2(B∗TD−1/2)†) = R(T∗B) = R(T∗), it holds that T?T∗is the
get?T∗T = χD(B∗T)B∗T and, therefore, using the same Lemma, R(?T∗) ⊆ N(B∗T)⊥D=
?
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As we have already mentioned in the previous section, {fn}n∈Nis a Riesz frames if and only if
N(T) is compatible with the canonical base (see Remark 3.4). If PD, Ndenote the (unique)
orthogonal projection onto the closed subspace N of ℓ2with respect to the inner product
?·, ·?D, it was proved in [1] that N is compatible if and only if
sup
D∈D(ℓ2)?PD, N? < ∞.
As a consequence of this result we obtain the following.
Theorem 4.5. Let F = {fn}n∈Nbe a frame for a closed subspace W of H, T its synthesis
operator, M another closed subspace of H such that H = W ˙ +M⊥and G = {gn}n∈N a
fixed (sampling) frame for M with synthesis operator B. Then, the following conditions are
equivalent:
1. F is a Riesz frame on W.
2. The oblique dual frames of T with respect to B that minimize the different weighted
norms are bounded from above. In other words
sup
D∈D(ℓ2)?B(D−1/2T∗B)†D−1/2? < ∞ .
Proof. Fix D ∈ D(ℓ2). We have already proved in Lemma 4.3 that
?
Hence
B(D−1/2T∗B)†D−1/2?∗T = T∗B(D−1/2T∗B)†D−1/2= T∗B χD(T∗B) = 1 − PD, N(T).
?B(D−1/2T∗B)†D−1/2? ≤ ?B? ?(D−1/2T∗B)†D−1/2?
= ?B? ?(T∗B)†(T∗B)(D−1/2T∗B)†D−1/2?
≤ ?B? ?(T∗B)†? ?(T∗B)(D−1/2T∗B)†D−1/2?
= ?B? ?(T∗B)†? ?1 − PD, N(T)? ,
and
?1 − PD, N(T)? = ?T∗B(D−1/2T∗B)†D−1/2? ≤ ?T∗? ?B(D−1/2T∗B)†D−1/2? .
Therefore
sup
D∈D(ℓ2)?B(D−1/2T∗B)†D−1/2? < ∞ ⇐⇒
sup
D∈D(ℓ2)?1 − PD, N(T)? < ∞ ,
which proves the Proposition.
?
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