- Open Access
Journal of Inequalities and Applications volume 2012, Article number: 196 (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.
Definition 1.1 A function is called Bochner measurable if there exists a sequence of simple functions such that
Definition 1.2 If μ is a measure on then X has the Radon-Nikodym property with respect to μ if for every countably additive vector measure γ on with values in X which has bounded variation and is absolutely continuous with respect to μ, there is a Bochner integrable function such that
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.
Remark 1.3 Suppose that is a measure space and has the Radon-Nikodym property. Let . The Bochner space of is defined to be the Banach space of (equivalence classes of) X-valued Bochner measurable functions F from Ω to X for which the norms
are finite. In [10, 11] and , p.51] it is proved that if and q is such that , then is isometrically isomorphic to if and only if has the Radon-Nikodym property. This isometric isomorphism is the mapping
where the mapping is defined on by
So for all and , we have
In the following, we use the notation instead of , so for all and ,
Particularly, if H is a Hilbert space, then is isometrically isomorphic to . So, for all and ,
in which does not mean the inner product of elements , in H, but
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.
Definition 2.1 Let . The family is a Bochner pg-frame for X with respect to if:
For each , is Bochner measurable,
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.
Example 2.2 Let be a frame for Hilbert space H, and μ be a counting measure on Ω. Set
Then is a Bochner pg-frame for H with respect to .
Example 2.3 Let , and be a measure such that , , and . Assume that and is a family of arbitrary Hilbert spaces and consider a fixed family such that , . Suppose that
It is clear that ’s are bounded and for each , is Bochner measurable. Also,
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.
Definition 2.4 Let be a Bochner pg-Bessel family for X with respect to , and q be the conjugate exponent of p. We define the operators T and U, by
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.
Proof Suppose that is a Bochner pg-Bessel family with bound B, and q is the conjugate exponent of p. We show that for all and all , the mapping is measurable. For all and , and G are Bochner measurable, so there are sequences of simple functions and such that
For each n, is a simple function and
and is measurable.
For each and , we have
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. .
Proof Let q be the conjugate exponent of p and . For all , we have
So , for all . There exists such that and
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.
Proof Let be a family of disjoint measurable subsets of Ω such that with for each . Let and assume, without loss of generality, , . Let
It is clear that for each , is measurable and , where is a family of disjoint and measurable subsets of Ω. If is not a Bochner pg-Bessel family for X, then there exists such that
and there exist finite sets I and J such that
Let be a sequence in V such that as . The assumption implies that
which is a contradiction to (2.4) (by the Lebesgue’s Dominated Convergence Theorem). So is a pg-Bessel family for X with respect to and Bessel bound B. Now, we show that
Since is a pg-Bessel family for X, the operator U defined by (2.3) is well defined and bounded. Assume that q is the conjugate exponent of p and let
It is obvious that for each , and G belong to , and we have
so by the Lebesgue’s Dominated Convergence Theorem,
Therefore, , hence
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.
Proof Let be a family of disjoint measurable subsets of Ω such that for each , and . Without loss of generality, we can assume , . Let
It is clear that for each , is measurable and , where is a family of disjoint and measurable subsets of Ω. We have
Suppose that , then there exists a family of disjoint finite subsets of such that
Let . Consider defined by
Then G is Bochner measurable, and
Therefore, . But
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.
Proof Let q be the conjugate exponent of p and for , consider
Then . So by Lemma 2.8. By Remark 1.3, and are isometrically isomorphic and . Therefore,
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.
Proof By assumption, there exist positive constants A and B such that
By Proposition 2.6, we have
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
(ii) Ifhas the lower Bochner pg-frame condition, then, where
are canonical mappings, and ψ is the mentioned isometrical isomorphism in Remark 1.3.
Proof (i) For each and , we have
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.
Proof (i) Since is a Bochner pg-frame, by Proposition 2.5, T is well defined and bounded. From the proof of Lemma 2.10, U is bounded below. So, by Lemma 1.4 and Lemma 2.12(i), is surjective.
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. □
Corollary 2.14 Ifis a Bochner pg-frame for X with respect toand q is the conjugate exponent of p, then for each, there existssuch that
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.
Proof From the proof of Theorem 2.9, for each , we have
By Proposition 2.6, ; consequently,
The operator is bounded below, so it has bounded inverse . We have
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.
Definition 3.1 Let . A family is called a Bochner qg-Riesz basis for with respect to , if:
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.
Proof (i) By Theorem 2.9 and Proposition 2.6 and Lemma 2.12 and Theorems 3.12, 4.7 and 4.12 in , it is obvious.
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.
Proof (i) → (ii): It is obvious.
→ (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.
Theorem 4.1 Suppose thatis a measure space where μ is σ-finite. Letbe a Bochner pg-frame for X with respect toand q be the conjugate exponent of p. Letbe a family such that for all, is Bochner measurable. Assume that there exist constants, , γ such that
for alland. Thenis a Bochner pg-frame for X with respect towith bounds
where A and B are the Bochner pg-frame bounds for.
Proof For each and , we have
Now, define by
so W is well defined and bounded. By Theorem 2.9, is a Bochner pg-Bessel family for X with bound .
Now, we show that satisfies the lower Bochner pg-frame condition. Let T and U be the synthesis and analysis operators of respectively. By Proposition 2.6, for all ,
By Lemma 2.10, is a closed subspace of , so is a bijective bounded operator, hence is alike. Since , so . Let and , then and , by Hahn-Banach theorem there exists such that and . It follows that
By Remark 1.3, there exists such that , then
Since , we have for each ,
From (4.1) and (4.3), we obtain that
For a given , there exists such that
Duffin RJ, Schaeffer AC: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 1952, 72: 341–366. 10.1090/S0002-9947-1952-0047179-6
Daubechies I, Grossmann A, Meyer Y: Painless nonorthogonal expansions. J. Math. Phys. 1986, 27(5):1271–1283. 10.1063/1.527388
Christensen O: Introduction to Frames and Riesz Bases. Birkhauser, Boston; 2003.
Abdollahpour MR, Faroughi MH, Rahimi A: PG -frames in Banach spaces. Methods Funct. Anal. Topol. 2007, 13: 201–210.
Casazza PG, Kutyniok G: Frame of subspaces. Contemp. Math. 2004, 345: 87–113.
Ali ST, Antoine JP, Gazeau JP: Continuous frames in Hilbert space. Ann. Phys. 1993, 222: 1–37. 10.1006/aphy.1993.1016
Rahimi A, Najati A, Dehghan YN: Continuous frames in Hilbert spaces. Methods Funct. Anal. Topol. 2006, 12(2):170–182.
Abdollahpour MR, Faroughi MH: Continuous G -frames in Hilbert spaces. Southeast Asian Bull. Math. 2008, 32: 1–19.
Cao HX, Li L, Chen QJ, Ji GX:-operator frames for a Banach space. J. Math. Anal. Appl. 2008, 347: 583–591. 10.1016/j.jmaa.2008.06.030
Diestel J, Uhl JJ Jr. Math. Surveys 15. In Vector Measure. Amer. Math. Soc., Providence; 1997.
Cengiz B: The dual of the Bochner space for arbitrary μ . Turk. J. Math. 1998, 22: 343–348.
Fleming RJ, Jamison JE Monographs and Surveys in Pure and Applied Mathematics 138. In Isometries on Banach Spaces: Vector-Valued Function Spaces (vol. 2). Chapman & Hall, London; 2008.
Murphy GJ: C∗-Algebras and Operator Theory. Academic Press, San Diego; 1990.
Heuser H: Functional Analysis. Wiley, New York; 1982.
Pedersen GK: Analysis Now. Springer, New York; 1998.
Megginson RE: An Introduction to Banach Space Theory. Springer, New York; 1998.
Rudin W: Functional Analysis. McGraw-Hill, New York; 1973.
The authors would like to sincerely thank Prof. Dr. Gitta Kutyniok and Dr. Asghar Ranjbari for their valuable comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
About this article
Cite this article
Faroughi, M.H., Rahmani, M. Bochner pg-frames. J Inequal Appl 2012, 196 (2012). https://doi.org/10.1186/1029-242X-2012-196
- Banach space
- Hilbert space
- Bochner measurable
- Bochner pg-frame
- Bochner pg-Bessel family
- Bochner qg-Riesz basis