© Faroughi and Rahmani; licensee Springer. 2012
Received: 10 May 2012
Accepted: 20 August 2012
Published: 4 September 2012
In this paper we introduce the concept of Bochner pg-frames for Banach spaces. We characterize the Bochner pg-frames and specify the optimal bounds of a Bochner pg-frame. Then we define a Bochner qg-Riesz basis and verify the relations between Bochner pg-frames and Bochner qg-Riesz bases. Finally, we discuss the perturbation of Bochner pg-frames.
1 Introduction and preliminaries
The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer  in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper  by Daubechies, Grossmann and Meyer, the frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames. Frames play a fundamental role in signal processing, image and data compression and sampling theory. They provided an alternative to orthonormal bases and have the advantage of possessing a certain degree of redundancy. A discrete frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into the expansion of frame elements. For more details about discrete frames, see . Resent results show that frames can provide a universal language in which many fundamental problems in pure mathematics can be formulated: the Kadison-Singer problem in operator algebras, the Bourgain-Tzafriri conjecture in Banach space theory, paving Toeplitz operators in harmonic analysis and many others. Various types of frames have been proposed, for example, pg-frames in Banach spaces , fusion frames , continuous frames in Hilbert spaces , continuous frames in Hilbert spaces , continuous g-frames in Hilbert spaces , -operator frames for a Banach space .
This paper is organized as follows. In Section 2, we introduce the concept of Bochner pg-frames for Banach spaces. Actually, continuous frames motivate us to introduce this kind of frames and analogous to continuous frames which are a generalized version of discrete frames, we want to generalize pg-frames in a continuous sense. Like continuous frames, these frames can be used in the areas where we need generalized frames in a continuous aspect. Also, we define corresponding operators (synthesis, analysis and frame operators) and discuss their characteristics and properties. In Section 3, we define a Bochner qg-Riesz basis and verify its relations by Bochner pg-frames. Finally, Section 4 is devoted to perturbation of Bochner pg-frames.
Throughout this paper, X and H will be a Banach space and a Hilbert space respectively, and is a family of Hilbert spaces.
Suppose that is a measure space, where μ is a positive measure.
The following definition introduces Bochner measurable functions.
for every set .
A Banach space X has the Radon-Nikodym property if X has the Radon-Nikodym property with respect to every finite measure. Spaces with Radon-Nikodym property include separable dual spaces and reflexive spaces, which include, in particular, Hilbert spaces.
where is the isometric isomorphism between H and , for more details refer to [, p.54].
We will use the following lemma which is proved in .
Lemma 1.4 Ifis a bounded operator from a Banach space X into a Banach space Y, then its adjointis surjective if and only if U has a bounded inverse on.
Note that for a collection of Hilbert spaces, we can suppose that there exists a Hilbert space K such that for all , , where is the direct sum of , see 3.1.5 in [, p.81].
2 Bochner pg-frames
Bochner spaces are often used in a functional analysis approach to the study of partial differential equations that depend on time, e.g., the heat equation: if the temperature is a scalar function of time and space, one can write to make f a function of time, with being a function of space, possibly in some Bochner space. Now, we intend to use this space to define a new kind of frames which contain all of continuous and discrete frames; in other words, we will generalize the g-frames to a continuous case that is constructed on the concept of Bochner spaces. Of course, this new frame can be useful in function spaces and operator theory to gain some general results that are achieved by g-frames or discrete frames.
2.1 Bochner pg-frames and corresponding operators
We start with the definition of Bochner pg-frames. Then we will give some characterizations of these frames.
For each , is Bochner measurable,
- (ii)there exist positive constants A and B such that(2.1)
A and B are called the lower and upper Bochner pg-frame bounds respectively. We call that is a tight Bochner pg-frame if A and B can be chosen such that , and a Parseval Bochner pg-frame if A and B can be chosen such that . If for each , , then is called a Bochner pg-frame for X with respect to H. A family is called a Bochner pg-Bessel family for X with respect to if the right inequality in (2.1) holds. In this case, B is called the Bessel bound.
Then is a Bochner pg-frame for H with respect to .
So, is a Bochner pg-frame for with respect to .
Now, we state the definition of some common corresponding operators for a Bochner pg-frame.
The operators T and U are called the synthesis and analysis operators of respectively.
The following proposition shows these operators are bounded. It is analogous to Theorem 3.2.3 in .
Proposition 2.5 Letbe a Bochner pg-Bessel family for X with respect toand with Bessel bound B. Then the operators T and U defined by (2.2) and (2.3) respectively, are well defined and bounded withand.
and is measurable.
Thus T is well defined, and . By a similar discussion, U is well defined and . □
The following proposition provides us with a concrete formula for the analysis operator.
Proposition 2.6 Ifis a Bochner pg-Bessel family for X with respect tothen for all, , a.e. .
which implies . Therefore, , a.e. . □
The following proposition shows that it is enough to check the Bochner pg-frame conditions on a dense subset. The discrete version of this proposition is available in [, Lemma 5.1.7].
Proposition 2.7 Suppose thatis a measure space where μ is σ-finite. Letbe a family such that for each, is Bochner measurable, and assume that there exist positive constants A and B such that (2.1) holds for all x in a dense subset V of X. Thenis a Bochner pg-frame for X with respect towith bounds A and B.
By letting , the proof is completed. □
2.2 Characterization of Bochner pg-frames
Now we give some characterizations of Bochner pg-frames in terms of their corresponding operators.
At first, we show the next lemma that is very useful in the case of complex valued -spaces.
Lemma 2.8 Letbe a measure space where μ is σ-finite. Letand q be its conjugate exponent. Ifis Bochner measurable and for each, , then.
which is a contradiction. □
The following theorem characterizes Bochner pg-Bessel families by the operator T defined by (2.2).
Theorem 2.9 Suppose thatis a measure space where μ is σ-finite. Letbe a family such that for eachthe mappingis Bochner measurable. If the operator T defined by (2.2) is well defined and bounded, thenis a Bochner pg-Bessel family for X with respect towith Bessel bound.
Similar to discrete frames, the analysis operator has closed range.
Lemma 2.10 Letbe a Bochner pg-frame for X with respect to. Then the operator U defined by (2.3) has closed range.
Hence U is bounded below. Therefore, U has closed range. □
The next proposition shows that there is no Bochner pg-frames for a non-reflexive Banach spaces.
Proposition 2.11 Letbe a Bochner pg-frame for X with respect to. Then X is reflexive.
Proof By Lemma 2.10, is a closed subspace of and is homeomorphism. Since is reflexive, so X is reflexive by Corollary 1.11.22 in . □
In the following lemma, we verify the adjoint operators of synthesis and analysis operators.
Lemma 2.12 Suppose thatis a Bochner pg-Bessel family for X with respect towith the synthesis operator T and the analysis operator U. Then
are canonical mappings, and ψ is the mentioned isometrical isomorphism in Remark 1.3.
- (ii)Since X and are reflexive, and are surjective. For each and ,
Hence . □
The following theorem characterizes Bochner pg-frames by the operator T defined by (2.2).
Theorem 2.13 Consider the family.
(i) Letbe a Bochner pg-frame for X with respect to. Then the operator T defined by (2.2) is a surjective bounded operator.
(ii) Letbe a measure space where μ is σ-finite and for each, be Bochner measurable. Let the operator T defined by (2.2) be a surjective bounded operator. Thenis a Bochner pg-frame for X with respect to.
- (ii)Since T is bounded, is a Bochner pg-Bessel family, by Theorem 2.9. Since is surjective, U has a bounded inverse on by Lemma 1.4. So there exists such that for all , . By Proposition 2.6, for all
Hence is a Bochner pg-frame. □
Proof It is obvious. □
The optimal Bochner pg-frame bounds can be expressed in terms of synthesis and analysis operators.
Theorem 2.15 Letbe a Bochner pg-frame for X with respect to. Thenandare the optimal upper and lower Bochner pg-frame bounds ofrespectively, whereis the inverse of U on, and T, U are the synthesis and analysis operators ofrespectively.
hence . □
3 Bochner qg-Riesz bases
In this section, we define Bochner qg-Riesz bases which are the generalization of Riesz bases and characterize their properties.
- (ii)for each , is Bochner measurable, and the operator T defined by (2.2) is well defined, and there are positive constants A and B such that
A and B are called the lower and upper Bochner qg-Riesz basis bounds of respectively.
Under some conditions, a Bochner qg-Riesz basis is a Bochner pg-frame, more precisely:
Proposition 3.2 Suppose thatis a measure space where μ is σ-finite and consider the family.
(i) Assume that for each, is Bochner measurable. is a Bochner qg-Riesz basis forwith respect toif and only if the operator T defined by (2.2) is an invertible bounded operator fromonto.
(ii) Letbe a Bochner qg-Riesz basis forwith respect towith the optimal upper Bochner qg-Riesz basis bound B. If p is the conjugate exponent of q, thenis a Bochner pg-frame for X with respect towith optimal upper Bochner pg-frame bound B.
By assumption and (i), the operator T defined by (2.2) is a bounded invertible operator. So by Theorem 2.13(ii), is a Bochner pg-frame for X with respect to with the optimal upper Bochner pg-frame bound B. □
The next theorem presents some equivalent conditions for a Bochner pg-frame being a Bochner qg-Riesz basis.
Theorem 3.3 Suppose thatis a measure space where μ is σ-finite. Letbe a Bochner pg-frame for X with respect towith the synthesis operator T and the analysis operator U, and q be the conjugate exponent of p. Then the following statements are equivalent:
(i) is a Bochner qg-Riesz basis for.
(ii) T is injective.
→ (i): By Theorem 2.13(i), the operator T defined by (2.2) is bounded and onto. By (ii), T is also injective. Therefore, T has a bounded inverse , and hence is a Bochner qg-Riesz basis for .
→ (iii): By Theorem 3.2, T is invertible, so is invertible. Lemma 2.12(ii) implies that .
→ (i): Since the operator U is invertible, by Lemma 2.12, is invertible. □
4 Perturbation of Bochner pg-frames
The perturbation of a discrete frame has been discussed in . In this section, we present another version of perturbation for Bochner pg-frames.
where A and B are the Bochner pg-frame bounds for.
so W is well defined and bounded. By Theorem 2.9, is a Bochner pg-Bessel family for X with bound .
The authors would like to sincerely thank Prof. Dr. Gitta Kutyniok and Dr. Asghar Ranjbari for their valuable comments.
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