International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 2 (2014), 148-158
http://www.etamaths.com

ON Λ-TYPE DUALITY OF FRAMES IN BANACH SPACES

RENU CHUGH1, MUKESH SINGH2, AND L. K. VASHISHT3,∗

Abstract. Frames are redundant system which are useful in the reconstruc-

tion of certain classes of spaces. The dual of a frame (Hilbert) always exists

and can be obtained in a natural way. Due to the presence of three Banach
spaces in the definition of retro Banach frames (or Banach frames) duality

of frames in Banach spaces is not similar to frames for Hilbert spaces. In

this paper we introduce the notion of Λ-type duality of retro Banach frames.
This can be generalized to Banach frames in Banach spaces. Necessary and

sufficient conditions for the existence of the dual of retro Banach frames are

obtained. A special class of retro Banach frames which always admit a dual
frame is discussed.

1. Introduction and Preliminaries

Let H be a Hilbert space with inner product 〈., .〉. A countable system E ≡
{fk} ⊂ H in a separable is called a frame (Hilbert) for H if there exists positive
constants A and B such that

A‖f‖2 ≤‖{〈f,fk〉}‖2`2 ≤ B‖f‖
2, for all f ∈H.(1.1)

The positive constants A and B are called the lower and upper frame bounds of the
frame E, respectively (the largest A and smallest B for which (1.1) holds are the
optimal frame bounds). They are not unique. A frame E ≡ {fn} for H is called
tight if it is possible to choose A = B and normalized tight if A = B = 1. If removal
of one fn renders the collection E ≡{fk} no longer a frame for H, then E is called
an exact frame for H. Let E ≡ {fk} be a frame(Hilbert) for H. The operator
T : `2 →H given by

T({ck}) =
∞∑
k=1

ckfk, {ck}∈ `2,

is called the synthesis operator or pre-frame operator of E.
Adjoint of T is the operator T∗ : H→ `2 given by

T∗(f) = {〈f,fk〉}

and is called the analysis operator of the frame E. Composing T and T∗ we obtain
the frame operator S = TT∗ : H→H given by

S(f) =

∞∑
k=1

〈f,fk〉fk,f ∈H.

2010 Mathematics Subject Classification. 42C15, 42C30; 46B15.

Key words and phrases. Frames, Banach frames, Retro Banach frames, Paley-Wiener space.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

148



ON DUALITY OF FRAMES IN BANACH SPACES 149

It can be easily verified that the frame operator S is a positive continuous invertible
linear operator from H onto H. Furthermore, every vector f ∈ H can be written
as:

f = SS−1f =

∞∑
k=1

〈S−1f,fk〉fk.(1.2)

The series given in (1.2) converges unconditionally for all f ∈ H and is called
frame decomposition or reconstruction formula for the frame. Thus, frames are
redundant building blocks which can recover the underlying space and which have
basis like property. One may observe that the frame decomposition shows that all
the information about a given signal (vector) f ∈ H is contained in the system
{〈S−1f,fk〉}. The scalars 〈S−1f,fk〉 are called the frame coefficients associated
with the frame E ≡{fk}.

Frames for Hilbert space were first introduced by Duffin and Schaeffer in [7],
while working in deep problem in non-harmonic Fourier series. For some reason the
work of Duffin and Schaeffer was not continued until 1986, when the fundamental
work of Daubechies, Grossmann and Mayer [6] brought this all back to life, right
at the dawn of the “wavelet era”. More precisely, Daubechies et al. in [6] observed
that frames can be used to find series expansions of functions in L2(R) which
are very similar to the expansions using orthonormal bases. This was probably
the time, when many mathematicians started to see the potential of the topic.
Since then, the theory of frames has been more widely studied. Gröchenig in [10],
generalized Hilbert frames to Banach spaces. Before the concept of Banach frames
was formalized, it appeared in the foundational work of Feichtinger and Gröchenig
[9] related to atomic decompositions. Atomic decompositions appeared in the field
of applied mathematics providing many applications. An atomic decomposition
allow a representation of every vector of the space via a series expansion in terms
of a fixed sequence of vectors which we call atoms. On the other hand Banach
frame for a Banach space ensure reconstruction via a bounded linear operator or
synthesis operator. In last one decade, various types of frames and reconstruction
system in Banach spaces have been introduced and studied. Retro Banach frames
were introduced and studied in [15] and further studied in [18, 19]. An excellent
approach towards the utility of frames in different direction is given in the book
by Casazza and Kutynoik [1] and in the paper by Heil and Walnut [12]. The basic
theory of frames can be found in [2, 3, 5, 11, 13, 21].

A warm up on duality of frames in Hilbert spaces: Let us give now a
brief discussion on duality of frames in Hilbert spaces. Suppose that E ≡{fk} is a
frame for a Hilbert space H. A system E′ ≡ {gk} ⊂ H is called a dual frame of E
if E′ is a frame for H and

f =

∞∑
k=1

〈f,gk〉fk, for all f ∈H.(1.3)

In short we say that (E,E′) is a dual pair. Let E ≡{fk} be a frame for H with frame
bounds 0 < A, B < ∞ and with frame operator S. Then E(natural) ≡ {S−1fk} is
a frame for H with frame bounds A−1,B−1. Furthermore, the condition in (1.3)
follows immediately follows from (1.2). Therefore, the system E(natural) ≡{S−1fk}
is a dual of E ≡ {fk}. The dual frame E(natural) ≡{S−1fk} is known as canonical
dual (or natural dual ) of E ≡{fk} . Thus, every frame for a Hilbert space admit a



150 CHUGH, SINGH, AND VASHISHT

dual (at least one). An interesting property of dual frames is that dual of a frame
need not be unique. In fact, there may be infinitely many duals for a non-exact
frame. A frame has a unique dual if and only if it is exact. For more technical
details about duality of frames (Hilbert), one may refer to [5, 13].
The following example shows that the dual of a frame (Hilbert) need not be unique.
Infact, it may have infinitely many duals.

Example 1.1. Let {χn} be an orthogonal basis for a Hilbert space H. Then, {χn}
is a Parseval frame for H. Consider a system E ≡ {fk} = {χ1,χ1,χ2,χ3,χ4, .....}.
One can easily verify that

‖f‖2 ≤‖{〈f,fk〉}‖2`2 ≤ 2‖f‖
2, for all f ∈H.

Therefore, E is a frame for H with one of the choice of bounds A = 1,B = 2. Let
S be the frame operator of E. Then, the canonical dual of E is given by

E′ ≡{S−1(fk)} = {
1

2
χ1,

1

2
χ1,χ2,χ2, ....}.

Other dual of E are E′′ ≡ {0,χ1, 0,χ2, .....} and E′′′ ≡ {13χ1,
2
3
χ1,χ2,χ3,χ4, .....}.

Therefore, infinitely many dual of a frame (Hilbert) can be constructed.

Motivation: Let us have a look on Example 1.1, where is observed that a vector
(signal) can be represented in infinitely many ways associated with a given frame.
Thus, there is an opportunity that even if the canonical dual frame is difficult to
find, there exists other duals that are easy to find. Now dual frames (like frames)
represents a vector in a concern space as series. On the other hand in case of retro
Banach frames a vector can be reconstructed by pre-frame operator(a bounded lin-
ear operator). More precisely, retro Banach frame recover the underlying space
via an operator and not by mean of an infinite series (see Definition 1.2). Due to
the presence of three Banach spaces in the definition of a retro Banach frame, it is
difficult to define its dual which preserve the property of reconstruction via infinite
series (as case of duals of frames for Hilbert spaces). Not only this, the dual of a
Hilbert frame has a relation with original frame that can be seen in (1.3). Some
author have work in the direction of dual of frame [4, 8, 11].

In this paper we introduce the notion of dual (or simply Λ-dual) of retro Banach
frames in Banach spaces. This can be generalized to Banach frames in Banach
spaces. It is observed that even an exact retro Banach frame does not admit a
dual retro Banach frame for the underlying space. The said situation is entirely
different from the dual of frames (Hilbert) for Hilbert spaces, where an exact frame
admit a unique dual. Necessary and sufficient conditions for the existence of retro
dual frames are obtained. Counterexamples are also given to defend results and
remarks. A special class of retro Banach frames which always admit retro dual
frames is discussed. In fact, we discussed an application of Young’s result given in
[20].

Now we give basic definitions and notations which will be required in this paper.
Throughout this paper X will denote an infinite dimensional Banach space over the
field K (K = R or C), X∗ the conjugate space of X and Zd is a Banach space of
scalar-valued sequences indexed by N which is associated with X∗. For a sequence
{Φn} in X , [Φn] denotes the closure of the linear hull of {Φn} in the norm topology
of X . As usual δk,m denote the Kronecker delta which is defined as δk,m = 0, if



ON DUALITY OF FRAMES IN BANACH SPACES 151

k 6= m and δk,m = 1, if k = m. The class of all bounded linear operator from a
Banach space X into a Banach space Y is denoted by B(X ,Y). Unless otherwise
stated all systems of the form {Φk} are indexed by N.

Definition 1.2. [18, at page 83] A system F ≡ ({Φk}, Θ) ({Φk} ⊂ X , Θ : Zd →
X∗) is called a retro Banach frame for X∗ with respect to an associated sequence
space Zd if,

(i) {Φ∗(Φk)}∈Zd, for each Φ∗ ∈X∗.
(ii) There exist positive constants (0 < A0 ≤ B0 < ∞) such that

A0‖Φ∗‖≤‖{Φ∗(Φk)}‖Zd ≤ B0‖Φ
∗‖, for each Φ∗ ∈X∗.

(iii) Θ is a bounded linear operator such that Θ({Φ∗(Φk)} = Φ∗, Φ∗ ∈X∗

The positive constant A0, B0 are called the lower and upper retro frame bounds
of ({Φk}∞k=1, Θ), respectively. As in case of frames (Hilbert) for Hilbert spaces,
they are not unique. The operator Θ : Zd → X∗ is called retro pre-frame operator
(or simply reconstruction operator) associated with {Φk}∞k=1. A retro Banach F ≡
({Φk}, Θ) is said to be exact if for each m ∈ N there exists no reconstruction
operator Θm such that ({Φk}, Θm)k 6=m is retro Banach frame for X∗. Retro Banach
frames were further studied in [18, 19].

Lemma 1.3. [18, Lemma 2 at page 83] Let F ≡ ({Φk}, Θ) be a retro Banach frame
for X∗ . Then, F is exact if and only if Φn 6∈ [Φk]k 6=n, for all n ∈ N.

Lemma 1.4. Let X be a Banach space and {Φ∗n} ⊂ X∗ be a sequence such that
{Φ ∈X : Φ∗n (Φ) = 0, for all n ∈ N} = {0}. Then, X is linearly isometric to the
Banach space Z = {{Φ∗n (Φ)} : Φ ∈X}, where the norm is given by

‖{Φ∗n(Φ)}‖Z = ‖Φ‖X , Φ ∈X .

2. Dual of Retro Banach Frames

Definition 2.1. Let F ≡ ({Φk}, Θ) be a retro Banach frame for X∗ and let Λ be
a fixed subset of N. A system {Φ∗k} ⊂ X

∗ is called dual retro Banach frame of F
(or simply Λ-dual of F) if

(i) Φ∗j (Φl) = δj,l, for all l ∈ N and for all j ∈ N\ Λ,
(ii) there exists a reconstruction operator Θ× such that G ≡ ({Φ∗k}, Θ

×) is a retro
Banach frame for X∗∗.

If G ≡ ({Φ∗k}, Θ
×) is a dual of F ≡ ({Φk}, Θ), then we say that F admits a dual

with respect to the system {Φ∗k}. In short we say that (F,G) is a retro dual pair.

Remark 2.2. In Definition 2.1, if Λ = ∅, then G is called the strong dual of F.

Remark 2.3. Recall that the dual of a frame (Hilbert) E for a Hilbert space H is
associated with E via series expansions (see equation (1.3)). The condition (i), in
Definition 2.1, gives a relation between a given retro Banach frame F and its dual
G. There may be other relation but at present we discuss Λ-type duality.

Example 2.4. Let {Φk} = {χk} be an orthogonal basis for a Hilbert space H.
Then

‖{Φ∗(Φk)}‖`2 = ‖Φ∗‖X∗, Φ∗ ∈X∗, for all Φ∗ ∈X∗.



152 CHUGH, SINGH, AND VASHISHT

Define Θ : `2 → X∗ by Θ({Φ∗(Φk)}) = Φ∗. Then, Θ is reconstruction operator
such that F ≡ ({Φk}, Θ) is a normalized tight exact retro Banach frame for X∗
with respect to Zd = `2.
Consider the system {Φ∗k}≡{χ

∗
1,χ
∗
1,χ
∗
2,χ
∗
3,χ
∗
4, .....}⊂H∗. Choose Λ = {1}⊂ N.

Then

Φ∗j (Φl) = δj,l, for all l ∈ N and for all j ∈ N\ Λ.

Furthermore, there exists a reconstruction operator Θ× such that G ≡ ({Φ∗k}, Θ
×)

is a retro Banach frame for X∗∗ with respect to Zd = `2 and with one of the choice
of bounds A = 1,B =

√
2. Hence (F,G) is a dual pair. It may be noted that

we can construct infinitely many dual retro Banach frames for F. More precisely,
there are infinitely many dual pair of the form (F,G).

Let F be an exact retro Banach frame for X∗. Then, in general, F has no retro
dual frame. The following example provides the existence of an exact retro Banach
frame F which does not admits a dual retro Banach frame.

Example 2.5. Consider the measure space X = L2(Ω) with counting measure,
where Ω = N. Let {χn} be an orthonormal basis of X .
Define {Φk}⊂X by

Φk = χk+1 + χ1, k ∈ N.
Let Z = {{Φ∗(Φk)} : Φ∗ ∈X∗}. Then, by using Lemma 1.4, Z is a Banach space
with norm given by

‖{Φ∗(Φk)}‖Z = ‖Φ∗‖X∗, Φ∗ ∈X∗.
Define Θ : Z →X∗ by

Θ({Φ∗(Φk)) = Φ∗, Φ∗ ∈X∗.
Then, Θ is a bounded linear operator such that F ≡ ({Φk}, Θ) is a retro Banach
frame for X∗ with respect to Z.

To show F is exact. Choose Φ∗k = χ
∗
k+1,k ∈ N. Then, {Φ

∗
k} ⊂ X

∗ and we
observe that

Φ∗n(Φm) = δn,m, for all n,m ∈ N.
Therefore, Φn 6∈ [Φk]k 6=n, for all n ∈ N. Thus, by using Lemma 1.3, we conclude
that F is an exact retro Banach frame for X∗. Note the if F is exact, then the
condition (i) given in definition 2.1 is satisfied for Λ = ∅.

To show that the condition (ii) given in Definition 2.1 is not satisfied. Let Θ×

be the reconstruction operator such that ({Φ∗k}, Θ
×) is a retro Banach frame for

X∗∗. Let A0, B0 be a choice of retro bounds for ({Φ∗k}, Θ
×).

Then

A0‖Φ∗∗‖≤‖{Φ∗∗(Φ∗k)}‖Zd ≤ B
0‖Φ∗‖, for all Φ∗∗ ∈X∗∗.(2.1)

In particular for Φ∗∗ = χ1, we have Φ
∗∗(Φ∗k) = 0, for all k ∈ N. Therefore, by using

retro frame inequality (2.1), we obtain Φ∗∗ = 0, a contradiction. Hence F has no
dual retro Banach frame. Furthermore, there is no other system {Ψ∗k} ⊂ X

∗ such
that F admits a dual with respect to {Ψ∗k}.

Remark 2.6. The second dual of a Banach space X may have a retro Banach
frame from its pre-dual space with respect to which a retro Banach frame for X∗
does not admit strong duality, but it may have a dual pair.



ON DUALITY OF FRAMES IN BANACH SPACES 153

The following example defend Remark 2.6

Example 2.7. Let F ≡ ({Φk}, Θ) be a retro Banach frame for X∗ given in Example
2.5. Choose Φ∗k = χ

∗
k,k ∈ N and let Z

0 = {{Φ∗∗(Φ∗k)} : Φ
∗∗ ∈X∗∗}. Then, Z0 is a

Banach space of sequences of scalars with norm given by

‖{Φ∗∗(Φ∗k)}‖Z0 = ‖Φ
∗∗‖X∗∗, Φ∗∗ ∈X∗∗.

Therefore, Θ× : {Φ∗∗(Φ∗k)}→ Φ
∗∗ is a bounded linear operator from Z0 onto X∗∗

such that G0 ≡ ({Φ∗k}, Θ
×) is a retro Banach frame for X∗∗ with respect to Z0 with

bounds A = B = 1. Choose Λ = {1}. Then, Φ∗j (Φl) = δj,l, for all l ∈ N and for all
j ∈ N\ Λ. Therefore, (F,G0) is a retro dual pair, which is not strong.

Remark 2.8. By Example 2.7 we observe that a retro Banach frame F for X∗
may have dual pair but does not admit a strong dual (even) with respect to an
orthonormal basis for X∗.

The following proposition provides sufficient conditions for the existence of the
dual retro Banach frames. It is sufficient to prove the result for the existence of
strong dual retro Banach frame. We can extend the same construction for the
existence of arbitrary dual pair.

Proposition 2.9. Let F ≡ ({Φk}, Θ) be a retro Banach frame for X∗. Then, F
admits a strong dual retro Banach frame if there exists a system ({Φ∗k}⊂X

∗ such
that Φ∗n(Φm) = δn,m, for all n,m ∈ N and there exists an injective closed linear
operator Φ∗∗ → {Φ∗∗(Φ∗k)} with closed range from X

∗∗ to Ẑ0, where Ẑ0 is some
Banach space of scalar valued sequences. These conditions are not necessary.

Proof. Let U : X∗∗ → Ẑ0 be given by U(Φ∗∗) = {Φ∗∗(Φ∗k)}, Φ
∗∗ ∈ X∗∗. Then,

by hypothesis, U is injective and closed linear operator with closed range R(U).

Therefore, U−1 : R(U) ⊂ Ẑ0 → X∗∗ is closed [14]. Thus, by using Closed Graph
Theorem [14], there exists a constant c > 0 such that

‖U(Φ∗∗)‖≥ c‖Φ∗∗‖, for all Φ∗∗ ∈X∗∗.

Let, if possible, there exists no reconstruction operator Θ× such that ({Φ∗k}, Θ
×) is

a retro Banach frame for X∗∗. Then, by using Hahn-Banach Theorem there exists
a non-zero functional Φ∗∗0 ∈X∗∗ such that Φ∗∗0 (Φ∗k) = 0, for all k ∈ N.
Therefore

0 = ‖U(Φ∗∗0 )‖≥ c‖Φ
∗∗
0 ‖.

This gives Φ∗∗0 = 0, a contradiction. Therefore, there exists a reconstruction opera-
tor Θ× such that ({Φ∗k}, Θ

×) is a retro Banach frame for X∗∗ with respect to some
associated Banach space of scalar valued sequences. Hence F admits a dual retro
Banach frame for the underlying space.

To show that the conditions are not necessary. Let X be the measure space given
in Example 2.5. Let Φk = k

2χk,k ∈ N, where {χk} is an orthonormal basis for X .
Then, there exists a bounded linear operator Θ such that F = ({Φk}, Θ) is a retro
Banach frame for X∗.
Choose Φ∗k =

1
k2
χk,k ∈ N. Then, Φ∗i (Φj) = δi,j for all i,j ∈ N. By the nature of

the system {Φ∗k}, we conclude that there exists a reconstruction operator Θ
× such

that G ≡ ({Φ∗k}, Θ
×) is a retro Banach frame for X∗∗ with respect to Ẑo = `2.

Thus, (F,G) is a dual pair.



154 CHUGH, SINGH, AND VASHISHT

Define Θ̃ : X∗∗ →Ẑo(= `2) by

Θ̃(Φ∗∗) = {Φ∗∗(Φ∗j )} = {
ξj
j2
}, Φ∗ = {ξj}∈X∗∗.

It can be verified that the range of the operator Θ̃ is not closed. �

The following theorem gives necessary and sufficient condition for a given retro
Banach frame to admit its dual.

Theorem 2.10. Let F ≡ ({Φn}, Θ) be a retro Banach frame for X∗. Then, F has
a dual retro Banach frame if and only if there exists a system {Φ∗n}⊂X∗ such that
Φ∗j (Φl) = δj,l, for all l ∈ N and for all j ∈ N \ Λ and dist(Φ

∗,Ln) → 0 as n →∞,
for all Φ∗ ∈ X∗, where Ln = [Φ∗1, Φ∗2, ...., Φ∗n], for all n ∈ N.

Proof. Suppose first that F ≡ ({Φn}, Θ) has a dual retro Banach frame. Then, by
definition we can find a system {Φ∗n} ⊂ X∗ such that Φ∗j (Φl) = δj,l, for all l ∈ N
and for all j ∈ N \ Λ and a reconstruction operator Θ× such that G ≡ ({Φ∗k}, Θ

×)
is a retro Banach frame for X∗∗. Let A0 and B0 be a choice of bounds for G.
Then

A0‖Φ∗∗‖X∗∗ ≤‖{Φ∗∗(Φ∗n)}‖(X∗∗)d ≤ B0‖Φ
∗∗‖X∗∗, for all Φ∗∗ ∈X∗∗.(2.2)

Suppose that the condition dist(Φ,Ln) → 0 as n → ∞, for all Φ∗ ∈ X∗, is not
satisfied. Then, there exists a non zero functional Φ∗0 ∈X∗ such that

lim
n→∞

dist(Φ∗0,Ln) 6= 0.

Note that dist(Φ∗0,Ln) ≥ dist(Φ∗0,Ln+1) for all n ∈ N. This is because {Ln} is a
nested system of subspaces and by definition of distance of a point from a set. Now
{dist(Φ∗0,Ln)} is bounded below monotone decreasing sequence. So, {dist(Φ∗0,Ln)}
is convergent and its limit is a positive real number, since otherwise dist(Φ∗0,Ln) →
0 as n →∞ (which is not possible). Let limn→∞ dist(Φ∗0,Ln) = ξ > 0.

Choose D =
⋃
n Ln. Then, by using the fact that dist(Φ

∗
0,D) = inf{dist(Φ∗0,Ln)},

we obtain

dist(Φ∗0,D) ≥ ξ > 0.(2.3)

Now we show that Φ∗0 6∈ D. Let if possible, Φ∗0 ∈ D. Then, we can find a
sequence {ζn}⊂ D such that dist(ζn, Φ∗0) → 0 as n →∞.

By using (2.3), we have dist(Φ∗0,D) ≥ ξ. Therefore, dist(ζn, Φ∗0) ≥ ξ > 0. This is
a contradiction to the fact that dist(ζn, Φ

∗
0) → 0 as n →∞. Hence Φ∗0 6∈ D. Thus,

by using Hahn-Banach theorem, there exists a non zero functional Φ∗∗0 ∈X∗∗ such
that

Φ∗∗0 (Φ
∗
n) = 0, for all n ∈ N.

Therefore, by using retro frame inequality (2.2), we have Φ∗∗0 = 0, a contradiction.
Hence dist(Φ∗,Ln) → 0 as n →∞, for all Φ∗ ∈ X∗.

To prove the converse part, assume that there exists a system {Φ∗n}⊂X∗ is such
that

Φ∗j (Φl) = δj,l, for all l ∈ N and for all j ∈ N\ Λ,

and

dist(Φ∗,Ln) → 0 as n →∞, for all Φ∗ ∈X∗.



ON DUALITY OF FRAMES IN BANACH SPACES 155

Then, in particular, for each � > 0 and for each Φ∗ ∈ X∗, we can find a Φ∗j from
some Lk such that

‖Φ∗ − Φ∗j‖ < �.

Therefore, by using Lemma 1.4, Z = {{Φ∗∗(Φ∗n)} : Φ∗∗ ∈ X∗∗} is a Banach space
of sequences of scalars with norm given by

‖{Φ∗∗(Φ∗n)}‖Z = ‖Φ
∗∗‖X∗∗, Φ∗∗ ∈X∗∗.

Define Θ0 : Z →X∗∗ by
Θ0({Φ∗∗(Φ∗k)}) = Φ

∗∗, Φ∗∗ ∈X∗∗.

Then, Θ0 is a bounded linear operator such that such that ({Φ∗n}, Θ0) is a retro
Banach frame for X∗∗ with respect to Z. Hence F has a dual retro Banach frame.

�

The existence of duality with respect to certain sequence space is also interested.
The following theorem provides the necessary and sufficient condition for an exact
retro Banach frame to admit a dual frame with respect to a given sequence space.

Theorem 2.11. Let F ≡ ({Φn}, Θ) be a retro Banach frame for X∗ with respect to
Zd and let Ad = {{Φ∗∗(Φ∗k)} : Φ

∗∗ ∈X∗∗}. Then, F has a dual retro Banach frame
with respect to the sequence space Ad if and only if there exists a system {Φ∗n}⊂X∗
such that Φ∗j (Φl) = δj,l, for all l ∈ N and for all j ∈ N\Λ and the analysis operator
U : Φ∗∗ →{Φ∗∗(Φ∗k)} is a bounded below continuous linear operator from X

∗∗ onto
Ad.

Proof. Suppose first that F has a dual retro Banach frame G with respect to Ad.
Then, there exists a reconstruction operator Θ× such that G ≡ ({Φ∗k}, Θ

×) is a retro
Banach frame for X∗∗ with respect to Ad. Therefore, there are positive constants
a0 and b0 such that

a0‖Φ∗∗‖≤‖{Φ∗∗(Φ∗k)}‖Ad ≤ b0‖Φ
∗∗‖, for each Φ∗∗ ∈X∗∗.(2.4)

Now consider the analysis operator U : X∗∗ →Ad which is given by
U(Φ∗∗) = {Φ∗∗(Φ∗k)}, Φ

∗∗ ∈X∗∗.
Then, linearity and ontoness of U is obvious. By using upper retro frame inequality
in (2.4), we have

‖U(Φ∗∗)‖Ad ≤ b0‖Φ
∗∗‖, for each Φ∗∗ ∈X∗∗.

Therefore, ‖U‖≤ b0. Hence U is continuous. Similarly, by using lower retro frame
inequality in (2.4), we have

‖U(•)‖Ad ≥ a0‖•‖X∗∗.
Hence U is bounded below.
For the reverse part, assume that U is bounded below. Then, using Lemma 1.4, Ad
is a Banach space with the norm given by

‖{Φ∗∗(Φ∗k)}‖Ad = ‖Φ
∗∗‖X∗∗, Φ∗∗ ∈X∗∗.

Define Θ× : Ad → X∗∗ by Θ×({Φ∗∗(Φ∗k)}) = Φ
∗∗. Then, Θ× is a bounded linear

operator such that ({Φ∗k}, Θ
×) is a retro Banach for X∗∗ with respect to Ad. The

theorem is proved. �



156 CHUGH, SINGH, AND VASHISHT

3. Applications

In this section we give some applications of the duality of retro Banach frames.
First two two examples provides an application of the Theorem 2.10.

Example 3.1. In this example, we give an application of Theorem 2.10 in duality
of retro Banach frames in finite dimensional Banach spaces. We start with a well
known frame in a finite dimensional Banach space, namely Mercedes-Benz frame.
Mercedes-Benz frame is one of the popular frame in finite dimensional frame theory.
It consists of three vectors which are equiangular situated on the unit circle. More
precisely, a Mercedes-Benz frame is a system consisting of three vectors in R2,
namely,

A =

{
Φ1 = (0, 1), Φ2 = (

−
√

3

2
,
−1
2

), Φ3 = (

√
3

2
,
−1
2

)

}
.

Let Zd0 = {[Φ∗(Φ1), Φ∗(Φ2), Φ∗(Φ3)]t : Φ ∈ X∗}. Then, Zd0 is a Banach space
with the norm given by

‖[Φ∗(Φ1), Φ∗(Φ2), Φ∗(Φ3)]t‖Zd0 = ‖Φ
∗‖X∗, Φ∗ ∈X∗.

Define Θ0 : Zd0 →X∗ by
Θ0([Φ

∗(Φ1), Φ
∗(Φ2), Φ

∗(Φ3)]
t) = Φ∗, Φ∗ ∈X∗.

Then, Θ0 is a bounded linear operator such that F0 ≡ ({Φi}3i=1, Θ0) is a retro
Banach frame for X∗ with respect to Zd0 and with bounds A = B = 1.

By using Theorem 2.10 and of above argument we conclude that every proper
subfamily of the Mercedes-Benz frame containing two vectors from A is a retro
Banach frame for (R2)∗ and admits a dual (strong) retro Banach frame for the
underlying space. This can be extended to arbitrary finite dimensional Banach
space. More precisely, every retro Banach frame for a finite dimensional space
contains a strong dual retro Banach frame for the underlying space.

For more applications of Mercedes-Benz frame an interested reader may refer to
[16, 17].

Example 3.2. Now we discuss Banach frames of wavelet system. Let X = L2(R).
Consider the Haar function Φ which is defined by

Φ(x) = 1, if x ∈ [0,
1

2
), Φ(x) = −1, if x ∈ [

1

2
, 1) and Φ(x) = 0, otherwise.

Let ψj,k(x) = 2
1
2 Φ(2jx−k),j,k ∈ Z and x ∈ R. The system {ψj,k(x)} is called a

wavelet system associated with the window function Φ. Without loss of generality,
we can write {Φn(•)} ≡ {ψj,k(•)}. Let Zd = {{ψ(Φn)} : ψ ∈ X∗}. Then, Zd is
Banach space of scalar valued sequences with the norm given by

‖{ψ(Φn)}‖Zd = ‖ψ‖X∗,ψ ∈X
∗.(3.1)

Define Θ : Zd →X∗ by
Θ({ψ(Φn)}) = ψ, ψ ∈X∗.

Then, Θ ∈ B(Zd,X∗). Hence Θ is a bounded linear operator such that F ≡
({Φn}, Θ) is a retro Banach frame for X∗ with bounds A = B = 1.

By the nature of construction of the system {Φn}, it can be easily verified that
Φn /∈ [Φm]m 6=n, for each n ∈ N.



ON DUALITY OF FRAMES IN BANACH SPACES 157

Thus, by Hahn-Banach theorem, there exists a {Φ∗n}⊂X∗ such that
Φ∗n(Φm) = δn,m, for all n,m ∈ N.

Hence F satisfying one of the sufficient conditions in Theorem 2.10, with Λ = ∅.
Let Ln = [Φ

∗
1, Φ

∗
2, ..., Φ

∗
n], for all n ∈ N. Note that each Φ∗j ∈ L

2(R) is given by

Φ∗j (Φ) =

∫
ΦΨΦdµ,

where ΨΦ is the representative of Φ.
Now by using retro frame inequality (3.1), we have

dist(Ψ∗,Ln) = inf
F∗∈Ln

‖Ψ∗ −F∗‖

= inf
F∗∈Ln

(∫
|Ψ∗ −F∗|2dµ

)1
2

→ 0 as n →∞, for all Ψ∗ ∈ X∗.
Hence by Theorem 2.10, F admits a dual (strong) retro Banach frame . We can
construct a dual of F which is not strong.

It is interested to know the class of retro Banach frames which always admits
dual retro Banach frames. It is difficult to characterize the class of retro Banach
frames which have dual retro Banach frames. A special class of retro Banach frames
consisting of complex exponentials in X = L2(−π,π) always admits a dual retro
Banach frame. This can be proved by using a result by Robert Young in [20 at page
538] . Recall that two sequences {Φn} and {Ψn} of elements from a Hilbert space
H are said to be biorthogonal if 〈Φn, Ψm〉 = δn,m for all n,m ∈ N. A sequence
that admits a biorthogonal sequence will be called minimal. A sequence {Φn}⊂H
is said to be complete in H if zero vector is alone is perpendicular to every Φn. A
sequence that is both minimal and complete will be called exact. Robert Young
considered a Paley-Wiener space consisting of all entire functions of exponential
type at most π that are square-integrable on the real axis and using Paley-Wiener
theorem proved the following result.

Theorem 3.3. [20, at page 538] If the sequence of complex exponential {eiλkt} is
exact in L2(−π,π), then its biorthogonal sequence is also exact.

To conclude the paper we show that retro Banach frames consisting of complex
exponentials in X = L2(−π,π) always admits a dual retro Banach frame.

Proposition 3.4. Let X = L2(−π,π) and Φk = eiλkt,k ∈ N. If F ≡ ({Φk}, Θ) be
an exact retro Banach frame for X∗ with admissible system {Φ∗k}⊂X

∗, then there
exists a reconstruction operator Θ× such that ({Φ∗k}, Θ

×) is a retro Banach frame
for X∗∗. More precisely, F has a dual (strong) retro Banach frame.

Proof. Suppose that there is no Θ× ∈ B(Zy,X∗∗) such that ({Φ∗k}, Θ
×) is a retro

Banach frame for X∗∗, where Zy is associated Banach space of scalar-valued se-
quences. Then, there exists a non-zero Φ∗∗ ∈X∗∗(≡X) such that

Φ∗∗(Φ∗k) =

∫ π
−π

Φ∗∗Φ∗kdt = 0, for all k ∈ N.

By using construction in the proof of Theorem 3.3 (for proof see [20] at page 538), we
can show that Φ∗∗ = 0. This is a contradiction. Hence there exists a reconstruction



158 CHUGH, SINGH, AND VASHISHT

operator Θ× ∈ B(Zy,X∗∗), where Zy = {{Φ∗∗(Φ∗k)} : Φ
∗∗ ∈X∗∗}, such that G ≡

({Φ∗k}, Θ
×) is a retro Banach frame for X∗∗. �

Acknowledgement

The third author was partially supported by R & D Doctoral Research Pro-
gramme, University of Delhi, Delhi. Letter No. Dean(R)/R&D/2012, dated July
03, 2012.

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1Department of Mathematics, Maharshi Dayanand University , Rohtak, Haryana,

India

2Department of Mathematics, Goverment College, Bahadurgarh, Maharshi Dayanand
University , Rohtak, Haryana, India

3Department of Mathematics, University of Delhi, Delhi-110007, India

∗Corresponding author