1. Introduction
Let
be an infinite dimensional separable complex Hilbert space with inner product
. A system
called a frame (Hilbert) for
if there exists positive constants A and B such that
![](https://www.scirp.org/html/3-5300429\694e5abd-a176-41c5-9dbe-d1cc1a3ac5dd.jpg)
The positive constants
and
are called lower and upper bounds of the frame
, respectively. They are not unique.
The operator
given by
is called the synthesis operator or pre-frame operator. Adjoint of T is given by
,
and is called the analysis operator. Composing
and
we obtain the frame operator
given by
. The frame operator S is a positive continuous invertible linear operator from
onto
. Every vector
can be written as:
![](https://www.scirp.org/html/3-5300429\e765b852-6fad-46bf-8fbd-f07a6ebfa6ee.jpg)
The series in the right hand side converge unconditionally and is called reconstruction formula for
. The representation of f in reconstruction formula need not be unique. Thus, frames are redundant systems in a Hilbert space which yield one natural representation for every vector in the concern Hilbert space, but which may have infinitely many different representations for a given vector.
Duffin and Schaeffer in [2] while working in nonharmonic Fourier series developed an abstract framework for the idea of time-frequency atomic decomposition by Gabor [3] and defined frames for Hilbert spaces. Due to some reason the theory of frames was not continued until 1986 when the fundamental work of Daubechies, Grossmann and Meyer published in [4]. Gröchenig in [5] generalized Hilbert frames to Banach spaces. Before the concept of Banach frames was formalized, it appeared in the foundational work of Feichtinger and Gröchenig [6,7] related to atomic decompositions. Atomic decompositions appeared in the field of applied mathematics providing many applications [8,9]. 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. Frames play an important role in the theory of nonuniform sampling [10], wavelet theory [11,12], signal processing [2,10], and many more. For a nice introduction of frames and their technical details one may refer to [13].
During the development of frames and expansions systems in Banach spaces Casazza and Christensen introduced reconstruction property for Banach spaces in [1]. Reconstruction property is an important tool in several areas of mathematics and engineering. In fact, it is related to bounded approximation property. Casazza and Christensen in [1] study perturbation theory related to reconstruction property. They develop more general perturbation theory that does not force equivalence of the sequences.
In this paper we introduce and study reconstruction property in Banach spaces which satisfy
-property. A characterization of
-reconstruction property in terms of frames in Banach spaces is obtained. Banach frames associated with reconstruction property are discussed.
2. Preliminaries
Throughout this paper
will denotes an infinite dimensional Banach space over a field
(which can be
or
),
be the conjugate space, and for a sequence
,
denotes closure of
in norm topology of
. The map
denotes the canonical mapping from
into
.
Definition 2.1 ([5]) Let
and
be given, where
is an associated Banach space of scalar valued sequences. A system
is called a Banach frame for
with respect to
if 1)
, for each
.
2) There exist positive constants C and D
such that
(2.1)
3)
is a bounded linear operator such that
![](https://www.scirp.org/html/3-5300429\872da2ab-2ad4-4bc7-9215-b9ab6774faa1.jpg)
As in case of frames for a Hilbert space, positive constants C and D are called lower and upper frame bounds of the Banach frame
, respectively. The operator
is called the reconstruction operator (or the pre-frame operator). The inequality 2.1 is called the frame inequality.
The Banach frame
is called tight if
and normalized tight if
. If there exists no reconstruction operator
such that ![](https://www.scirp.org/html/3-5300429\ad9e2702-77f6-4fcb-a6a3-d311170cec9e.jpg)
is Banach frame for
, then
will be called an exact Banach frame.
The notion of retro Banach frames introduced and studied in [14].
Definition 2.2 ([14]) A system ![](https://www.scirp.org/html/3-5300429\4c5d528c-852b-4d52-999d-2ee1d3a9f9d3.jpg)
is called a retro Banach frame for
with respect to an associated sequence space
if 1)
, for each
.
2) There exist positive constants
such that
![](https://www.scirp.org/html/3-5300429\942ba9d1-47c2-4202-bbcc-c4fada0d88c1.jpg)
3)
is a bounded linear operator from
onto
.
The positive constant
are called retro frame bounds of
and operator
is called retro pre-frame operator (or simply reconstruction operator) associated with
.
Lemma 2.3. Let
be a Banach space and
be a sequence such that ![](https://www.scirp.org/html/3-5300429\43df59de-5727-4986-a261-a96a0695a436.jpg)
. Then,
is linearly isometric to the Banach space
, where the norm is given by
.
Casazza and Christensen in [1] introduced reconstruction property in Banach spaces.
Definition 2.4 ([1]) Let
be a separable Banach space. We say that a sequence
has the reconstruction property for
with respect to
, if
![](https://www.scirp.org/html/3-5300429\e38b5359-cd11-46b1-a0df-9399bb7cb9cb.jpg)
In short, we will also say
has reconstruction property for
. More precisely, we say that
is a reconstruction system for
.
Remark 2.5 An interesting example for a reconstruction property is given in [1]: Let
and ![](https://www.scirp.org/html/3-5300429\bb5008a4-4fee-4cf8-8ea1-06cda7cd2b63.jpg)
is unitarily equivalent to the unit vector basis of
.
Then,
has a reconstruction property with respect to its own pre-dual (that is, expansions with respect to the orthonormal basis). Further examples on reconstruction property are discussed in Example 3.4.
Definition 2.6 A reconstruction system
for
is said to be 1) pre-shrinking if
.
2) shrinking if
is a reconstruction system for
.
Regarding existence of Banach spaces which have reconstruction system, Casazza and Christensen proved the following result.
Proposition 2.7 ([1]) There exists a Banach space
with the following properties:
1) There is a sequence
such that each ![](https://www.scirp.org/html/3-5300429\1775d9f2-26d2-41ec-99f7-c021cf050e41.jpg)
has a expansion ![](https://www.scirp.org/html/3-5300429\bda4030d-f163-49d5-875a-4c7b9168650a.jpg)
2)
does not have the reconstruction property with respect to any pair ![](https://www.scirp.org/html/3-5300429\ef9cf07d-508f-489c-bb12-95f304ef8d61.jpg)
The notion of reconstruction property is related to Bounded Approximation Property (BAP). If ![](https://www.scirp.org/html/3-5300429\5e4446be-9e1c-4400-9f3d-581355834b99.jpg)
has reconstruction property for
, then
has the bounded approximation property. So,
is isomorphic to a complemented subspace of a Banach space with a basis. It is also used to study geometry of Banach spaces. For more results and basics on reconstruction property and bounded approximation property one may refer to [15] and references therein.
3.
-Reconstruction Property
Definition 3.1 Suppose
has the reconstruction property for
with respect to
.
Then, we say that
satisfy property
if
and there exists a functional ![](https://www.scirp.org/html/3-5300429\c71f2561-673a-4dc8-8893-9cc1dc4d0105.jpg)
such that
, for all
. In this case we say that
is a
-reconstruction system for
.
Remark 3.2 If
and there exists a functional
such that
, for all
then we say that
is a
-reconstruction system (or weak
-reconstruction system for
).
Remark 3.3 A
-reconstruction system is actually a dual system of a
-Schauder frame [16] in the context of reconstruction property.
Example 3.4 Let
and
be a sequence of canonical unit vectors. Define
by
.
Then,
has a reconstruction property with respect to
, where
Hence
is a
-reconstruction system for
[See Proposition 3.5]. Note that the reconstruction system
is shrinking.
Now define
by
. Then, ![](https://www.scirp.org/html/3-5300429\b9100328-65ea-4b25-a523-dee4ca54a7e0.jpg)
has a reconstruction property with respect to
, where
By Proposition 3.5,
is not a
-reconstruction system for
.
Note that
is
-reconstruction system which is shrinking. Thus, a shrinking reconstruction system for
need not be a
-reconstruction system.
We now give a characterization of a
-reconstruction system for
as claimed in section 1, in terms of frames.
Proposition 3.5 Let
be a reconstruction system for
with
. Then, ![](https://www.scirp.org/html/3-5300429\c0a7595d-b193-4965-8a49-71b640af7f28.jpg)
satisfy property
if and only if there is no retro preframe operator
such that
is retro Banach frame for
.
This is an immediate consequence of the following lemma.
Lemma 3.6 Let
be a pre-shrinking reconstruction system for
. Then,
is a
-reconstruction system if and only if there exists no retro pre-frame operator
such that
is retro Banach frame for
.
Proof. Forward part is obvious. Indeed, by using lower retro frame inequality of
and existence of
such that
for all ![](https://www.scirp.org/html/3-5300429\de5ccf34-045e-42ba-9d6e-fe0217a66405.jpg)
we obtain
This is a contradiction.
For reverse part, let if possible, there is no reconstruction operator
such that
is a retro Banach frame for
. Then, Hahn Banach Theorem force to admit a non zero functional
such that
, for all
. That is,
, for all
. Put
, for all
. If
then
for all
But
is pre-shrinking, therefore
, a contradiction. Thus
. Put
. Then,
is such that
for all
Thus,
is a
-reconstruction system. ![](https://www.scirp.org/html/3-5300429\5644132c-5172-426d-a177-0624ce2ca4dc.jpg)
Remark 3.7 Note that Lemma 3.6 is no longer true if
is not pre-shrinking.
Application: Let
. Consider a boundary value problem(BVP) with a set of n boundary conditions:
BVP: ![](https://www.scirp.org/html/3-5300429\780d2a82-21c2-4386-a8a5-07d9f6479cb1.jpg)
where
is a linear differential operator with
and
denotes the set of n boundary conditions:
![](https://www.scirp.org/html/3-5300429\89311f45-72cb-4d20-bcb7-1696fa66dc89.jpg)
It is given in [17] (at page 66) that for a large class of boundary conditions (which are known as regular boundary conditions), the BVP admits a system
and
consisting of eigenfunction associated with given BVP such that
![](https://www.scirp.org/html/3-5300429\97ad53c9-ccde-48b1-bffa-a81c43ceea60.jpg)
![](https://www.scirp.org/html/3-5300429\42168a46-01b0-4f7c-a79d-1ac9605ec7ea.jpg)
It is well known that the corresponding to
there exists a ![](https://www.scirp.org/html/3-5300429\a9245a43-1e45-43a8-8e93-9c14ae275d97.jpg)
such that
is a reconstruction system for
Now
and
![](https://www.scirp.org/html/3-5300429\f58c3f59-4352-4d3a-8be9-08bd279200c9.jpg)
Therefore, by using Paley and Wiener theorem in [18, p. 208], there exists a sequence
such that
admits a reconstruction system with respect to
. This reconstruction system is not of type
. Therefore, by using Lemma 3.6, there exists a retro pre-frame operator
such that
is retro Banach frame for
. Recall that if we write a function in terms of reconstruction system, then computation of all the coefficients is required. If calculation of coefficients which appear in the series expansion of a given reconstruction system are complicated, then we reconstruct the function by pre-frame operator of
.
The following proposition provides a sufficient condition for a reconstruction system to satisfy property
.
Proposition 3.8 Let
be a reconstruction system for
. If there exists a vector
in
such that
for all
, then
is a
-reconstruction system.
Proof. Let
be the canonical embedding of
into
. Then
is such that
, for all
. Thus,
is a
reconstruction system for
.
Remark 3.9 The condition in Proposition 3.8 is not necessary. However, if
is reflexive, then the condition given in Proposition 3.8 turns out to be necessary. Moreover, this is equivalent to the condition: There exists no pre-frame operator
such that
is a Banach frame for
.
To conclude the section we show that a given
- reconstruction system in Banach spaces produce another
-reconstruction system: Consider a
-reconstruction system
for
.
Let ![](https://www.scirp.org/html/3-5300429\3ebe824c-1e20-4c9e-8b70-bdaee5a5d9d3.jpg)
Then
is a Banach space with norm given by
![](https://www.scirp.org/html/3-5300429\2ae06cc7-82f4-4e4c-b34e-18520c4a3e38.jpg)
Define
by
.
Then
is an isomorphism of
into ![](https://www.scirp.org/html/3-5300429\c5ca0709-4520-4a34-b81f-2eb496a21a1c.jpg)
Also
defined by
is also a bounded linear operator from
onto
.
Put
. Then
is a closed subspace of
such that
Moreover, if
is any element such that
, then
and
![](https://www.scirp.org/html/3-5300429\ddb33335-3435-4e96-8398-94f74943a618.jpg)
Therefore,
is such that
![](https://www.scirp.org/html/3-5300429\4b4d69b8-5899-4533-90c9-e705b0ef525f.jpg)
Hence ![](https://www.scirp.org/html/3-5300429\f2e16840-2121-40e8-90ea-d6fa58b5ae82.jpg)
Let V be projection on
onto
.
Then,
. Thereforefor each
, we have
![](https://www.scirp.org/html/3-5300429\b342d877-11e0-4062-8f7e-dadb4970b78e.jpg)
That is:
for all
So,
for all
, where
is sequence of canonical unit vectors in
. Hence
is a reconstruction system for ![](https://www.scirp.org/html/3-5300429\7092163a-28fc-4938-b766-b2cab15de471.jpg)
which satisfy property
.
This is summarized in the following proposition.
Proposition 3.10 Let
be a
-reconstruction system for
. Then, there exists
such that
is a
-reconstruction system for
, where
and
are same as in above discussion.
4. Associated Banach Frames
Definition 4.1 Suppose that
has the reconstruction property for
with respect to
. Then, there exists a reconstruction operator
such that
is a Banach frame for
with respect to some
. We say that
is an associated Banach frame of
.
Consider a reconstruction system
for a Banach space
. We can write each element of
(we can reconstruct
) by mean of an infinite series formed by
over scalars
. For a non zero functional
(say), in general, there is
• no
such that
has the reconstruction property for
with respect to
.
• no reconstruction operator
such that
is a Banach frame for
.
More precisely, two natural and important problem arise, namely, existence of
such that
has the reconstruction property for
with respect to
and other is the existence of a reconstruction operator
associated with
. Cassaza and Christensen in [1] study some stability of reconstruction property in Banach spaces in terms of closeness of certain sequence to a given reconstruction system. In the present section we focus on pre-frame operator associated with
.
Motivation: Consider a signal space
. If
is a frame (Hilbert) for
, then each element of
can be recovered by an infinite combinations of frame elements. That is, by the reconstruction formula. If a signal f is transmitted to a receiver, then there are some kind of disturbances in the received signal. To overcome these disturbances from the receiver, frames plays an important role. Actually, a signal in the space (after its transmission) is in the form of the frame coefficients
,
. An error
is always is expected with concern signal in the space. That is, actual signal in the space is of the form
, where
is an error associated with f. An interesting discussion in this direction is given in [13]. We extend the said problem to Banach frames in general Banach spaces.
The following proposition provides sufficient condition for a reconstruction system to satisfy property
in terms of non-existence of pre-frame operator associated with certain error.
Proposition 4.2 Suppose that
has the reconstruction property for a signal space (Banach)
with respect to
. Let
(error) be in
for which there is no pre-frame operator
such that
is a Banach frame for
, then
is a
-reconstruction system for
.
Proof. Let
be an associated Banach frame of
. If there exists no pre-frame operator ![](https://www.scirp.org/html/3-5300429\2469a62f-5fe5-468c-bd48-f7ef10eeb2c4.jpg)
such that
is a Banach frame for ![](https://www.scirp.org/html/3-5300429\b59c873a-136e-49ed-ada8-59cdf061b7ad.jpg)
then, there is a non-zero vector
such that
, for all
. By frame inequality of
, we conclude that
. Put
. Then,
is such that
, for all
. Hence
is a
-reconstruction system for
.
Remark 4.3 The condition in Proposition 4.2 is not necessary unless
correspond to a vector in
. More precisely, we can find a certain error
such that there exists no pre-frame operator
associated with
provided
.
Remark 4.4 Let us continue with the outcomes in Proposition 4.2, where
is found to be a
-reconstruction system for
provided there is no pre-frame operator
such that
is a Banach frame for
, where
is certain choice of error (functional). A natural problem arises, which is of determining a Banach space
for which the system
admits a pre-frame operator. Answer to this problem is positive, provided
is preshrinking. The outline of construction of such a Banach space can be understood as follows: Put
(where
is same as in the proof of Proposition 4.2). Now, there is no pre-frame operator
associated with
, so there exists a nonzero vector
such that
, for all
. By using frame inequality of the associated Banach frame
we have
. Put
. Then,
is a non-zero vector in
such that
, for all
. Therefore,
for all
. Now
is pre-shrinking, so we have
. Hence
, where
. By using Lemma 2.3 there exists a pre-frame operator
such that
is a Banach frame(normalized tight) for the Banach space
, where
;
.
An application of Proposition 4.2 is given below:
Example 4.5 Let
be a reconstruction system given in Example 3.4 for
. Then,
is a bounded linear operator such that
is a Banach frame (associated) for
with respect to
and with bounds
. Put
(this choice makes sense, because disturbances are not constant!). Then,
is an error in
for which there is no reconstruction operator
such that
is a Banach frame for
. Hence by Proposition 4.2,
is a
-reconstruction system for
. ![](https://www.scirp.org/html/3-5300429\896e36bb-b71f-4449-8da3-4636f6114fd1.jpg)
Definition 4.6 Fix
. A pair
, (where
) is said to be localized at
, if
, where
is a sequence of scalars.
If
is localized at every
with
for all
, then
turns out to be a reconstruction system for
. Consider a reconstruction system
for
and
be its associated Banach frame with respect to
. Let
. Then, in general, there is no pre-frame operator
associated with system
. This problem is also known as stability of
with respect to
. If
is not localized at certain vectors in
then we can find such pre-frame operator associated with
. This is what concluding proposition of this paper says.
Proposition 4.7 Let
be a reconstruction system for
. Assume that
is not localized at
, where
.
Then, there exists a pre-frame operator
, such that
is a Banach frame for
.
Proof. Let
be associated Banach frame of
. Let, if possible, there is no reconstruction operator
, such that
is a Banach frame for
. Then, there exists a non zero vector
such that
for all ![](https://www.scirp.org/html/3-5300429\7352311c-c0f8-4bc3-91dd-e61e91029e47.jpg)
This gives
![](https://www.scirp.org/html/3-5300429\699cf14f-1ab1-4fd7-9c8a-d34c67dbac11.jpg)
By using frame inequality of
, we obtain,
![](https://www.scirp.org/html/3-5300429\a844a5f3-b397-4670-afcd-ecb6fa5642ad.jpg)
Since
is a reconstruction system for
, we have
![](https://www.scirp.org/html/3-5300429\d69449b4-2c45-4b7c-8051-637752bc5513.jpg)
Thus,
is localized at
, where
, a contradiction. Hence there exists a preframe operator
, such that
is a Banach frame for
.
5. Conclusion
The notion of
-reconstruction property is proposed in section 3 and its characterization in terms of frames in Banach spaces is given. More precisely, Proposition 3.5 characterize
-reconstruction property in terms of existence of pre-frame operator but in a contrapositive way. This situation is same as in electrodynamics, where there is a game of movement of electron but charge given to electron is negative! Moreover, the action of a functional from
on a given system from
decide the existence of pre-frame operator associated with certain system. This looks like dynamics of reconstruction property. By motivation from the theory of frames for Hilbert spaces which control the perturbed system associated with a signal in space(after its transmission), we extend the said situation to Banach spaces. More precisely, Proposition 4.2 control the situation in abstract setting via non-existence of pre-frame operator. Finally, the notion of local reconstruction system is proposed and its utility in complicated stability of associated Banach frames is reflected in Proposition 4.7.