Bochner pg-frames

Open AccessResearch

DOI: 10.1186/1029-242X-2012-196

Cite this article as:
Faroughi, M.H. & Rahmani, M. J Inequal Appl (2012) 2012: 196. doi:10.1186/1029-242X-2012-196

Abstract

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.

MSC:42C15, 46G10.

Keywords

Banach spaceHilbert spaceframeBochner measurableBochner pg-frameBochner pg-Bessel familyBochner qg-Riesz basis

1 Introduction and preliminaries

The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper [2] 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 [3]. 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 [4], fusion frames [5], continuous frames in Hilbert spaces [6], continuous frames in Hilbert spaces [7], continuous g-frames in Hilbert spaces [8], ( p , Y ) Open image in new window-operator frames for a Banach space [9].

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 { H ω } ω Ω Open image in new window is a family of Hilbert spaces.

Suppose that ( Ω , Σ , μ ) Open image in new window is a measure space, where μ is a positive measure.

The following definition introduces Bochner measurable functions.

Definition 1.1 A function f : Ω X Open image in new window is called Bochner measurable if there exists a sequence of simple functions { f n } n = 1 Open image in new window such that
lim n f n ( ω ) f ( ω ) = 0 , a.e.  [ μ ] . Open image in new window
Definition 1.2 If μ is a measure on ( Ω , Σ ) Open image in new window then X has the Radon-Nikodym property with respect to μ if for every countably additive vector measure γ on ( Ω , Σ ) Open image in new window with values in X which has bounded variation and is absolutely continuous with respect to μ, there is a Bochner integrable function g : Ω X Open image in new window such that
γ ( E ) = E g ( ω ) d μ ( ω ) Open image in new window

for every set E Σ Open image in new window.

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 ( Ω , Σ , μ ) Open image in new window is a measure space and X Open image in new window has the Radon-Nikodym property. Let 1 p Open image in new window. The Bochner space of L p ( μ , X ) Open image in new window 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
F p = ( Ω F ( ω ) p d μ ( ω ) ) 1 p , 1 p < , F = ess sup ω Ω F ( ω ) , p = Open image in new window
are finite. In [10, 11] and [12], p.51] it is proved that if 1 p < Open image in new window and q is such that 1 p + 1 q = 1 Open image in new window, then L q ( μ , X ) Open image in new window is isometrically isomorphic to ( L p ( μ , X ) ) Open image in new window if and only if X Open image in new window has the Radon-Nikodym property. This isometric isomorphism is the mapping
ψ : L q ( μ , X ) ( L p ( μ , X ) ) , g ψ ( g ) , Open image in new window
where the mapping ψ ( g ) Open image in new window is defined on L p ( μ , X ) Open image in new window by
ψ ( g ) ( f ) = Ω g ( ω ) ( f ( ω ) ) d μ ( ω ) , f L p ( μ , X ) . Open image in new window
So for all f L p ( μ , X ) Open image in new window and g L q ( μ , X ) Open image in new window, we have
f , ψ ( g ) = Ω f ( ω ) , g ( ω ) d μ ( ω ) . Open image in new window
In the following, we use the notation f , g Open image in new window instead of f , ψ ( g ) Open image in new window, so for all f L p ( μ , X ) Open image in new window and g L q ( μ , X ) Open image in new window,
f , g = Ω f ( ω ) , g ( ω ) d μ ( ω ) . Open image in new window
Particularly, if H is a Hilbert space, then ( L p ( μ , H ) ) Open image in new window is isometrically isomorphic to L q ( μ , H ) Open image in new window. So, for all f L p ( μ , H ) Open image in new window and g L q ( μ , H ) Open image in new window,
f , g = Ω f ( ω ) , g ( ω ) d μ ( ω ) , Open image in new window
in which f ( ω ) , g ( ω ) Open image in new window does not mean the inner product of elements f ( ω ) Open image in new window, g ( ω ) Open image in new window in H, but
f ( ω ) , g ( ω ) = ν ( g ( ω ) ) ( f ( ω ) ) , Open image in new window

where ν : H H Open image in new window is the isometric isomorphism between H and H Open image in new window, for more details refer to [[13], p.54].

We will use the following lemma which is proved in [14].

Lemma 1.4If U : X Y Open image in new windowis a bounded operator from a Banach spaceXinto a Banach spaceY, then its adjoint U : Y X Open image in new windowis surjective if and only ifUhas a bounded inverse on R U Open image in new window.

Note that for a collection { H β } β B Open image in new window of Hilbert spaces, we can suppose that there exists a Hilbert space K such that for all β B Open image in new window, H β K Open image in new window, where K = β B H β Open image in new window is the direct sum of { H β } β B Open image in new window, see 3.1.5 in [[15], 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 g ( t , x ) Open image in new window is a scalar function of time and space, one can write ( f ( t ) ) ( x ) : = g ( t , x ) Open image in new window to make f a function of time, with f ( t ) Open image in new window 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 1 < p < Open image in new window. The family { Λ ω B ( X , H ω ) : ω Ω } Open image in new window is a Bochner pg-frame for X with respect to { H ω } ω Ω Open image in new window if:
  1. (i)

    For each x X Open image in new window, ω Λ ω ( x ) Open image in new window is Bochner measurable,

     
  2. (ii)
    there exist positive constants A and B such that
    A x ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p B x , x X . Open image in new window
    (2.1)
     

A and B are called the lower and upper Bochner pg-frame bounds respectively. We call that { Λ ω } ω Ω Open image in new window is a tight Bochner pg-frame if A and B can be chosen such that A = B Open image in new window, and a Parseval Bochner pg-frame if A and B can be chosen such that A = B = 1 Open image in new window. If for each ω Ω Open image in new window, H ω = H Open image in new window, then { Λ ω } ω Ω Open image in new window is called a Bochner pg-frame for X with respect to H. A family { Λ ω B ( X , H ω ) : ω Ω } Open image in new window is called a Bochner pg-Bessel family for X with respect to { H ω } ω Ω Open image in new window if the right inequality in (2.1) holds. In this case, B is called the Bessel bound.

Example 2.2 Let { f i } i I Open image in new window be a frame for Hilbert space H, Ω = I Open image in new window and μ be a counting measure on Ω. Set
Λ i : H C , Λ i ( h ) = h , f i , h H . Open image in new window

Then { Λ i } i I Open image in new window is a Bochner pg-frame for H with respect to C Open image in new window.

Example 2.3 Let Ω = { a , b , c } Open image in new window, Σ = { , { a , b } , { c } , Ω } Open image in new window and μ : Σ [ 0 , ] Open image in new window be a measure such that μ ( ) = 0 Open image in new window, μ ( { a , b } ) = 1 Open image in new window, μ ( { c } ) = 1 Open image in new window and μ ( Ω ) = 2 Open image in new window. Assume that X = L p ( Ω ) Open image in new window and { H ω } ω Ω Open image in new window is a family of arbitrary Hilbert spaces and consider a fixed family { h ω } ω Ω { H ω } ω Ω Open image in new window such that h ω = 1 Open image in new window, ω Ω Open image in new window. Suppose that
Λ ω : L p ( Ω ) H ω , Λ ω ( φ ) = φ ( c ) h ω . Open image in new window
It is clear that Λ ω Open image in new window’s are bounded and for each φ L p ( Ω ) Open image in new window, ω Λ ω ( φ ) Open image in new window is Bochner measurable. Also,
Ω Λ ω ( x ) p d μ ( ω ) = Ω | φ ( c ) | p d μ ( ω ) = | φ ( c ) | p μ ( Ω ) = 2 | φ ( c ) | p . Open image in new window

So, { Λ ω } ω Ω Open image in new window is a Bochner pg-frame for L p ( Ω ) Open image in new window with respect to { H ω } ω Ω Open image in new window.

Now, we state the definition of some common corresponding operators for a Bochner pg-frame.

Definition 2.4 Let { Λ ω } ω Ω Open image in new window be a Bochner pg-Bessel family for X with respect to { H ω } ω Ω Open image in new window, 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 { Λ ω } ω Ω Open image in new window respectively.

The following proposition shows these operators are bounded. It is analogous to Theorem 3.2.3 in [3].

Proposition 2.5Let { Λ ω B ( X , H ω ) : ω Ω } Open image in new windowbe a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω Open image in new windowand with Bessel boundB. Then the operatorsTandUdefined by (2.2) and (2.3) respectively, are well defined and bounded with T B Open image in new windowand U B Open image in new window.

Proof Suppose that { Λ ω } ω Ω Open image in new window is a Bochner pg-Bessel family with bound B, and q is the conjugate exponent of p. We show that for all x X Open image in new window and all G L q ( μ , ω Ω H ω ) Open image in new window, the mapping ω Λ ω ( x ) , G ( ω ) Open image in new window is measurable. For all x X Open image in new window and G L q ( μ , ω Ω H ω ) Open image in new window, ω Λ ω ( x ) Open image in new window and G are Bochner measurable, so there are sequences of simple functions { λ n } n = 1 Open image in new window and { g n } n = 1 Open image in new window such that
lim n λ n ( ω ) Λ ω ( x ) = 0 , a.e.  [ μ ] , lim n g n ( ω ) G ( ω ) = 0 , a.e.  [ μ ] . Open image in new window
For each n, λ n , g n Open image in new window is a simple function and
| Λ ω ( x ) , G ( ω ) λ n ( ω ) , g n ( ω ) | | Λ ω ( x ) λ n ( ω ) , G ( ω ) | + | λ n ( ω ) , g n ( ω ) G ( ω ) | Λ ω ( x ) λ n ( ω ) G ( ω ) + λ n ( ω ) g n ( ω ) G ( ω ) . Open image in new window
So
lim n | Λ ω ( x ) , G ( ω ) λ n ( ω ) , g n ( ω ) | = 0 Open image in new window

and ω Λ ω ( x ) , G ( ω ) Open image in new window is measurable.

For each x X Open image in new window and G L q ( μ , ω Ω H ω ) Open image in new window, we have
| x , T G | = | Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) | Ω Λ ω x G ( ω ) d μ ( ω ) ( Ω Λ ω x p d μ ( ω ) ) 1 p ( Ω G ( ω ) q d μ ( ω ) ) 1 q B x G q . Open image in new window

Thus T is well defined, and T B Open image in new window. By a similar discussion, U is well defined and U B Open image in new window. □

The following proposition provides us with a concrete formula for the analysis operator.

Proposition 2.6If { Λ ω } ω Ω Open image in new windowis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω Open image in new windowthen for all x X Open image in new window, ( U x ) ( ω ) = Λ ω x Open image in new window, a.e. [ μ ] Open image in new window.

Proof Let q be the conjugate exponent of p and x X Open image in new window. For all G L q ( μ , ω Ω H ω ) Open image in new window, we have
U x , G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) = { Λ ω x } ω Ω , G . Open image in new window
So U x { Λ ω x } ω Ω , G = 0 Open image in new window, for all G L q ( μ , ω Ω H ω ) Open image in new window. There exists G L q ( μ , ω Ω H ω ) Open image in new window such that G q = 1 Open image in new window and
U x { Λ ω x } ω Ω , G = U x { Λ ω x } ω Ω p , Open image in new window

which implies U x { Λ ω x } ω Ω p = 0 Open image in new window. Therefore, ( U x ) ( ω ) = Λ ω x Open image in new window, a.e. [ μ ] Open image in new window. □

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 [[3], Lemma 5.1.7].

Proposition 2.7Suppose that ( Ω , Σ , μ ) Open image in new windowis a measure space whereμisσ-finite. Let { Λ ω B ( X , H ω ) : ω Ω } Open image in new windowbe a family such that for each x X Open image in new window, ω Λ ω ( x ) Open image in new windowis Bochner measurable, and assume that there exist positive constantsAandBsuch that (2.1) holds for allxin a dense subsetVofX. Then { Λ ω } ω Ω Open image in new windowis a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowwith boundsAandB.

Proof Let { Ω n } n = 1 Open image in new window be a family of disjoint measurable subsets of Ω such that Ω = n = 1 Ω n Open image in new window with μ ( Ω n ) < Open image in new window for each n 1 Open image in new window. Let x X Open image in new window and assume, without loss of generality, Λ ω x 0 Open image in new window, ω Ω Open image in new window. Let
Δ m x = { ω Ω | m 1 < Λ ω x m } , m = 0 , 1 , 2 , . Open image in new window
It is clear that for each m = 0 , 1 , 2 , Open image in new window , Δ m x Ω Open image in new window is measurable and Ω = m = 0 , n = 1 ( Δ m x Ω n ) Open image in new window, where { Δ m x Ω n } n = 1 , m = 0 Open image in new window is a family of disjoint and measurable subsets of Ω. If { Λ ω } ω Ω Open image in new window is not a Bochner pg-Bessel family for X, then there exists x X Open image in new window such that
( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p > B x . Open image in new window
So
m , n Δ m x Ω n Λ ω ( x ) p d μ ( ω ) > B p x p Open image in new window
and there exist finite sets I and J such that
m I n J Δ m x Ω n Λ ω ( x ) p d μ ( ω ) > B p x p . Open image in new window
(2.4)
Let { x k } k = 1 Open image in new window be a sequence in V such that x k x Open image in new window as k Open image in new window. The assumption implies that
m I n J Δ m x Ω n Λ ω ( x k ) p d μ ( ω ) B p x k p , Open image in new window
which is a contradiction to (2.4) (by the Lebesgue’s Dominated Convergence Theorem). So { Λ ω } ω Ω Open image in new window is a pg-Bessel family for X with respect to { H ω } ω Ω Open image in new window and Bessel bound B. Now, we show that
( Ω Λ ω ( x k ) p d μ ( ω ) ) 1 p ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p . Open image in new window
Since { Λ ω } ω Ω Open image in new window 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
G : Ω ω Ω H ω , G ( ω ) = Λ ω x p q q Λ ω x , ω Ω , Open image in new window
and
G k : Ω ω Ω H ω , G k ( ω ) = Λ ω x k p q q Λ ω x k , ω Ω . Open image in new window
It is obvious that for each k N Open image in new window, G k Open image in new window and G belong to L q ( μ , ω Ω H ω ) Open image in new window, and we have
U x , G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) = Ω Λ ω ( x ) p d μ ( ω ) , U x k , G k = Ω Λ ω ( x k ) , G k ( ω ) d μ ( ω ) = Ω Λ ω ( x k ) p d μ ( ω ) . Open image in new window
Since lim k ( G k G ) ( ω ) q = 0 Open image in new window and
( G k G ) ( ω ) q ( G k ( ω ) + G ( ω ) ) q 2 q 1 ( G k ( ω ) q + G ( ω ) q ) , Open image in new window
so by the Lebesgue’s Dominated Convergence Theorem,
lim k Ω ( G k G ) ( ω ) q d μ ( ω ) = 0 . Open image in new window
Therefore, lim k G k G q = 0 Open image in new window, hence
| Ω Λ ω ( x k ) p d μ ( ω ) Ω Λ ω ( x ) p d μ ( ω ) | = | U x k , G k U x , G | | Ω Λ ω ( x k x ) , G k ( ω ) d μ ( ω ) | + | Ω Λ ω ( x ) , ( G k G ) ( ω ) d μ ( ω ) | ( Ω Λ ω ( x k x ) p d μ ( ω ) ) 1 p ( Ω G k ( ω ) q d μ ( ω ) ) 1 q + ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p ( Ω ( G G k ) ( ω ) q d μ ( ω ) ) 1 q B x k x G k q + B x | G k G q . Open image in new window

By letting x k x Open image in new window, 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 L p Open image in new window-spaces.

Lemma 2.8Let ( Ω , Σ , μ ) Open image in new windowbe a measure space whereμisσ-finite. Let 1 < p < Open image in new windowandqbe its conjugate exponent. If F : Ω H Open image in new windowis Bochner measurable and for each G L q ( μ , H ) Open image in new window, | Ω F ( ω ) , G ( ω ) H d μ ( ω ) | < Open image in new window, then F L p ( μ , H ) Open image in new window.

Proof Let { Ω n } n = 1 Open image in new window be a family of disjoint measurable subsets of Ω such that for each n 1 Open image in new window, μ ( Ω n ) < Open image in new window and Ω = n = 1 Ω n Open image in new window. Without loss of generality, we can assume F ( ω ) 0 Open image in new window, ω Ω Open image in new window. Let
Δ m = { ω Ω | m 1 < F ( ω ) m } , m = 0 , 1 , 2 , . Open image in new window
It is clear that for each m = 0 , 1 , 2 , Open image in new window , Δ m Ω Open image in new window is measurable and Ω = m = 0 , n = 1 ( Δ m Ω n ) Open image in new window, where { Δ m Ω n } n = 1 , m = 0 Open image in new window is a family of disjoint and measurable subsets of Ω. We have
Ω F ( ω ) p d μ ( ω ) = m = 0 n = 1 Δ m Ω n F ( ω ) p d μ ( ω ) Open image in new window
and
Δ m Ω n F ( ω ) p d μ ( ω ) m p μ ( Ω n ) < . Open image in new window
Suppose that Ω F ( ω ) p d μ ( ω ) = Open image in new window, then there exists a family { E k } k = 1 Open image in new window of disjoint finite subsets of N 0 × N Open image in new window such that
( m , n ) E k Δ m Ω n F ( ω ) p d μ ( ω ) > 1 . Open image in new window
Let E = k = 1 ( m , n ) E k ( Δ m Ω n ) Open image in new window. Consider G : Ω H Open image in new window defined by
G ( ω ) = { c k p q F ( ω ) p q q F ( ω ) if  ω ( m , n ) E k ( Δ m Ω n ) , k = 1 , 2 , , 0 if  ω Ω E , Open image in new window
where
c k : = 1 k q p ( ( m , n ) E k ( Δ m Ω n ) F ( ω ) p d μ ( ω ) ) 1 p . Open image in new window
Then G is Bochner measurable, and
Ω G ( ω ) q d μ ( ω ) = E G ( ω ) q d μ ( ω ) = k = 1 ( m , n ) E k Δ m Ω n G ( ω ) q d μ ( ω ) = k = 1 ( m , n ) E k ( Δ m Ω n ) c k p F ( ω ) p d μ ( ω ) = k = 1 1 k q < . Open image in new window
Therefore, G L q ( μ , H ) Open image in new window. But
| Ω F ( ω ) , G ( ω ) H d μ ( ω ) | = k = 1 c k p q ( m , n ) E k ( Δ m Ω n ) F ( ω ) p d μ ( ω ) = k = 1 1 k ( ( m , n ) E k ( Δ m Ω n ) F ( ω ) p d μ ( ω ) ) 1 p > k = 1 1 k = , Open image in new window

which is a contradiction. □

The following theorem characterizes Bochner pg-Bessel families by the operator T defined by (2.2).

Theorem 2.9Suppose that ( Ω , Σ , μ ) Open image in new windowis a measure space whereμisσ-finite. Let { Λ ω B ( X , H ω ) : ω Ω } Open image in new windowbe a family such that for each x X Open image in new windowthe mapping ω Λ ω ( x ) Open image in new windowis Bochner measurable. If the operatorTdefined by (2.2) is well defined and bounded, then { Λ ω } ω Ω Open image in new windowis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω Open image in new windowwith Bessel bound T Open image in new window.

Proof Let q be the conjugate exponent of p and for x X Open image in new window, consider
F x : L q ( μ , ω Ω H ω ) C , F x ( G ) = x , T G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) , G L q ( μ , ω Ω H ω ) . Open image in new window
Then F x ( L q ( μ , ω Ω H ω ) ) Open image in new window. So { Λ ω x } ω Ω L p ( μ , ω Ω H ω ) Open image in new window by Lemma 2.8. By Remark 1.3, ( L q ( μ , ω Ω H ω ) ) Open image in new window and L p ( μ , ω Ω H ω ) Open image in new window are isometrically isomorphic and { Λ ω x } ω Ω p = F x Open image in new window. Therefore,
( Ω Λ ω x p d μ ( ω ) ) 1 p = F x = sup G q = 1 | x , T G | T x . Open image in new window

 □

Similar to discrete frames, the analysis operator has closed range.

Lemma 2.10Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new window. Then the operatorUdefined by (2.3) has closed range.

Proof By assumption, there exist positive constants A and B such that
A x ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p B x , x X . Open image in new window
By Proposition 2.6, we have
A x U x p B x . Open image in new window

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 2.11Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new window. ThenXis reflexive.

Proof By Lemma 2.10, R U Open image in new window is a closed subspace of L p ( μ , ω Ω H ω ) Open image in new window and U : X R U Open image in new window is homeomorphism. Since L p ( μ , ω Ω H ω ) Open image in new window is reflexive, so X is reflexive by Corollary 1.11.22 in [16]. □

In the following lemma, we verify the adjoint operators of synthesis and analysis operators.

Lemma 2.12Suppose that { Λ ω } ω Ω Open image in new windowis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω Open image in new windowwith the synthesis operatorTand the analysis operatorU. Then

(i) U = T Open image in new window.

(ii) If { Λ ω } ω Ω Open image in new windowhas the lower Bochnerpg-frame condition, then T J 1 = ψ J 2 U Open image in new window, where
J 1 : X X Open image in new window
and
J 2 : L p ( μ , ω Ω H ω ) ( L p ( μ , ω Ω H ω ) ) Open image in new window

are canonical mappings, andψis the mentioned isometrical isomorphism in Remark  1.3.

Proof (i) For each G L q ( μ , ω Ω H ω ) Open image in new window and x X Open image in new window, we have
U x , G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) = x , T G , Open image in new window
so U = T Open image in new window.
  1. (ii)
    Since X and L p ( μ , ω Ω H ω ) Open image in new window are reflexive, J 1 Open image in new window and J 2 Open image in new window are surjective. For each G L q ( μ , ω Ω H ω ) Open image in new window and x X Open image in new window,
    G , T J 1 x = T G , J 1 x = x , T G = U x , G , Open image in new window
     
also
G , ψ J 2 U x = ψ G , J 2 U x = U x , ψ G = U x , G . Open image in new window

Hence T J 1 = ψ J 2 U Open image in new window. □

The following theorem characterizes Bochner pg-frames by the operator T defined by (2.2).

Theorem 2.13Consider the family { Λ ω B ( X , H ω ) : ω Ω } Open image in new window.

(i) Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new window. Then the operatorTdefined by (2.2) is a surjective bounded operator.

(ii) Let ( Ω , Σ , μ ) Open image in new windowbe a measure space whereμisσ-finite and for each x X Open image in new window, ω Λ ω ( x ) Open image in new windowbe Bochner measurable. Let the operatorTdefined by (2.2) be a surjective bounded operator. Then { Λ ω } ω Ω Open image in new windowis a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new window.

Proof (i) Since { Λ ω } ω Ω Open image in new window 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), U = T Open image in new window is surjective.
  1. (ii)
    Since T is bounded, { Λ ω } ω Ω Open image in new window is a Bochner pg-Bessel family, by Theorem 2.9. Since T = U Open image in new window is surjective, U has a bounded inverse on R U Open image in new window by Lemma 1.4. So there exists A > 0 Open image in new window such that for all x X Open image in new window, U x p A x Open image in new window. By Proposition 2.6, for all x X Open image in new window
    A x U x p = ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p . Open image in new window
     

Hence { Λ ω } ω Ω Open image in new window is a Bochner pg-frame. □

Corollary 2.14If { Λ ω } ω Ω Open image in new windowis a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowandqis the conjugate exponent ofp, then for each x X Open image in new window, there exists G L q ( μ , ω Ω H ω ) Open image in new windowsuch that
x , x = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) , x X . Open image in new window

Proof It is obvious. □

The optimal Bochner pg-frame bounds can be expressed in terms of synthesis and analysis operators.

Theorem 2.15Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new window. Then T Open image in new windowand U ˜ Open image in new windoware the optimal upper and lower Bochnerpg-frame bounds of { Λ ω } ω Ω Open image in new windowrespectively, where U ˜ Open image in new windowis the inverse ofUon R U Open image in new window, andT, Uare the synthesis and analysis operators of { Λ ω } ω Ω Open image in new windowrespectively.

Proof From the proof of Theorem 2.9, for each x X Open image in new window, we have
( Ω Λ ω x p d μ ( ω ) ) 1 p = F x = sup G q = 1 | x , T G | . Open image in new window
Therefore,
sup x = 1 ( Ω Λ ω x p d μ ( ω ) ) 1 p = sup x = 1 F x = sup x = 1 sup G q = 1 | x , T G | = sup G q = 1 sup x = 1 | x , T G | = sup G q = 1 T G = T . Open image in new window
By Proposition 2.6, U x p = ( Ω Λ ω x p d μ ( ω ) ) 1 p Open image in new window; consequently,
inf x = 1 U x p = inf x = 1 ( Ω Λ ω x p d μ ( ω ) ) 1 p . Open image in new window
The operator U : X L p ( μ , ω Ω H ω ) Open image in new window is bounded below, so it has bounded inverse U ˜ : R U X Open image in new window. We have
inf x = 1 U x p = inf x 0 U x p x = inf U y ˜ 0 y p U y ˜ = inf y 0 y p U y ˜ = 1 sup y 0 U y ˜ y p = 1 U ˜ , Open image in new window

hence inf x = 1 ( Ω Λ ω x p d μ ( ω ) ) 1 p = 1 U ˜ Open image in new window. □

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 1 < q < Open image in new window. A family { Λ ω B ( X , H ω ) : ω Ω } Open image in new window is called a Bochner qg-Riesz basis for X Open image in new window with respect to { H ω } ω Ω Open image in new window, if:
  1. (i)

    { x : Λ ω x = 0 ,  a.e.  [ μ ] } = { 0 } Open image in new window,

     
  2. (ii)
    for each x X Open image in new window, ω Λ ω ( x ) Open image in new window 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 G q T G B G q , G L q ( μ , ω Ω H ω ) . Open image in new window
     

A and B are called the lower and upper Bochner qg-Riesz basis bounds of { Λ ω } ω Ω Open image in new window respectively.

Under some conditions, a Bochner qg-Riesz basis is a Bochner pg-frame, more precisely:

Proposition 3.2Suppose that ( Ω , Σ , μ ) Open image in new windowis a measure space whereμisσ-finite and consider the family { Λ ω B ( X , H ω ) : ω Ω } Open image in new window.

(i) Assume that for each x X Open image in new window, ω Λ ω ( x ) Open image in new windowis Bochner measurable. { Λ ω } ω Ω Open image in new windowis a Bochnerqg-Riesz basis for X Open image in new windowwith respect to { H ω } ω Ω Open image in new windowif and only if the operatorTdefined by (2.2) is an invertible bounded operator from L q ( μ , ω Ω H ω ) Open image in new windowonto X Open image in new window.

(ii) Let { Λ ω } ω Ω Open image in new windowbe a Bochnerqg-Riesz basis for X Open image in new windowwith respect to { H ω } ω Ω Open image in new windowwith the optimal upper Bochnerqg-Riesz basis boundB. Ifpis the conjugate exponent ofq, then { Λ ω } ω Ω Open image in new windowis a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowwith optimal upper Bochnerpg-frame boundB.

Proof (i) By Theorem 2.9 and Proposition 2.6 and Lemma 2.12 and Theorems 3.12, 4.7 and 4.12 in [17], it is obvious.
  1. (ii)

    By assumption and (i), the operator T defined by (2.2) is a bounded invertible operator. So by Theorem 2.13(ii), { Λ ω } ω Ω Open image in new window is a Bochner pg-frame for X with respect to { H ω } ω Ω Open image in new window 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 3.3Suppose that ( Ω , Σ , μ ) Open image in new windowis a measure space whereμisσ-finite. Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowwith the synthesis operatorTand the analysis operatorU, andqbe the conjugate exponent ofp. Then the following statements are equivalent:

(i) { Λ ω } ω Ω Open image in new windowis a Bochnerqg-Riesz basis for X Open image in new window.

(ii) Tis injective.

(iii) R U = L p ( μ , ω Ω H ω ) Open image in new window.

Proof (i) → (ii): It is obvious.
  1. (ii)

    → (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 T 1 : X L q ( μ , ω Ω H ω ) Open image in new window, and hence { Λ ω } ω Ω Open image in new window is a Bochner qg-Riesz basis for X Open image in new window.

     
  2. (i)

    → (iii): By Theorem 3.2, T is invertible, so T Open image in new window is invertible. Lemma 2.12(ii) implies that R U = L p ( μ , ω Ω H ω ) Open image in new window.

     
  3. (iii)

    → (i): Since the operator U is invertible, by Lemma 2.12, T = U Open image in new window is invertible. □

     

4 Perturbation of Bochner pg-frames

The perturbation of a discrete frame has been discussed in [3]. In this section, we present another version of perturbation for Bochner pg-frames.

Theorem 4.1Suppose that ( Ω , Σ , μ ) Open image in new windowis a measure space whereμisσ-finite. Let { Λ ω } ω Ω Open image in new windowbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowandqbe the conjugate exponent of p. Let { Θ ω B ( X , H ω ) : ω Ω } Open image in new windowbe a family such that for all x X Open image in new window, ω Θ ω ( x ) Open image in new windowis Bochner measurable. Assume that there exist constants λ 1 Open image in new window, λ 2 Open image in new window, γsuch that
0 λ 2 < 1 , λ 2 λ 1 < 1 , 0 γ < ( 1 λ 1 2 λ 2 ) A Open image in new window
for all G L q ( μ , ω Ω H ω ) Open image in new windowand x X Open image in new window. Then { Θ ω } ω Ω Open image in new windowis a Bochnerpg-frame forXwith respect to { H ω } ω Ω Open image in new windowwith bounds
A [ ( 1 λ 1 2 λ 2 ) γ A 1 λ 2 ] and B [ 1 + λ 1 + γ B 1 λ 2 ] , Open image in new window

whereAandBare the Bochnerpg-frame bounds for { Λ ω } ω Ω Open image in new window.

Proof For each x X Open image in new window and G L q ( μ , ω Ω H ω ) Open image in new window, we have
| Ω Θ ω x , G ( ω ) d μ ( ω ) | | Ω ( Λ ω Θ ω ) x , G ( ω ) d μ ( ω ) | + | Ω Λ ω x , G ( ω ) d μ ( ω ) | ( 1 + λ 1 ) | Ω Λ ω x , G ( ω ) d μ ( ω ) | + λ 2 | Ω Θ ω x , G ( ω ) d μ ( ω ) | + γ G q . Open image in new window
So
| Ω Θ ω x , G ( ω ) d μ ( ω ) | 1 + λ 1 1 λ 2 | Ω Λ ω x , G ( ω ) d μ ( ω ) | + γ 1 λ 2 G q 1 + λ 1 1 λ 2 B G q x + γ 1 λ 2 G q = [ 1 + λ 1 1 λ 2 B x + γ 1 λ 2 ] G q . Open image in new window
Now, define W : L q ( μ , ω Ω H ω ) X Open image in new window by
x , W G = Ω Θ ω x , G ( ω ) d μ ( ω ) , x X , G L q ( μ , ω Ω H ω ) . Open image in new window
Since
W G = sup x = 1 | x , W G | = sup x = 1 | Ω Θ ω x , G ( ω ) d μ ( ω ) | [ 1 + λ 1 1 λ 2 B + γ 1 λ 2 ] G q , Open image in new window

so W is well defined and bounded. By Theorem 2.9, { Θ ω } ω Ω Open image in new window is a Bochner pg-Bessel family for X with bound B [ 1 + λ 1 + γ B 1 λ 2 ] Open image in new window.

Now, we show that { Θ ω } ω Ω Open image in new window satisfies the lower Bochner pg-frame condition. Let T and U be the synthesis and analysis operators of { Λ ω } ω Ω Open image in new window respectively. By Proposition 2.6, for all x X Open image in new window,
A x U x p B x . Open image in new window
By Lemma 2.10, R U Open image in new window is a closed subspace of L p ( μ , ω Ω H ω ) Open image in new window, so Q = U : X R U Open image in new window is a bijective bounded operator, hence ( Q 1 ) : X R U Open image in new window is alike. Since B 1 Q 1 A 1 Open image in new window, so ( Q 1 ) = Q 1 A 1 Open image in new window. Let x X Open image in new window and S = ( Q 1 ) Open image in new window, then S A 1 Open image in new window and S ( x ) R U Open image in new window, by Hahn-Banach theorem there exists φ L p ( μ , ω Ω H ω ) Open image in new window such that φ | R U = S ( x ) Open image in new window and φ = S ( x ) Open image in new window. It follows that
φ = S ( x ) S x A 1 x . Open image in new window
(4.2)
By Remark 1.3, there exists G L q ( μ , ω Ω H ω ) Open image in new window such that ψ ( G ) = φ Open image in new window, then
G q = φ A 1 x . Open image in new window
(4.3)
Since x = Q ( Q 1 ) ( x ) Open image in new window, we have for each x X Open image in new window,
x , x = x , ( Q S ) ( x ) = U x , S ( x ) = U x , φ = U x , ψ ( G ) = Ω Λ ω x , G ( ω ) d μ ( ω ) . Open image in new window
From (4.1) and (4.3), we obtain that
x W G = sup x = 1 | x , x W G | = sup x = 1 | Ω ( Λ ω Θ ω ) x , G ( ω ) μ ( ω ) | sup x = 1 [ λ 1 | x , x | + λ 2 | x , W G | + γ G q ] sup x = 1 [ ( λ 1 + λ 2 ) | x , x | + λ 2 | x , W G x | + γ A 1 x ] sup x = 1 [ ( λ 1 + λ 2 ) x x + λ 2 W G x x + γ A 1 x ] ( λ 1 + λ 2 + γ A 1 ) x + λ 2 W G x , Open image in new window
which implies
W G x ( λ 1 + λ 2 ) A + γ ( 1 λ 2 ) A x . Open image in new window
For a given x X Open image in new window, there exists x X Open image in new window such that
x = 1 , x = x ( x ) . Open image in new window
Hence
x = x ( x ) = | x , x | = | x , x W G + x , W G | | x , x W G | + | x , W G | ( λ 1 + λ 2 ) A + γ ( 1 λ 2 ) A x x + | Ω Θ ω x , G ( ω ) d μ ( ω ) | ( λ 1 + λ 2 ) A + γ ( 1 λ 2 ) A x x + ( Ω Θ ω x p d μ ( ω ) ) 1 p G q ( λ 1 + λ 2 ) A + γ ( 1 λ 2 ) A x x + A 1 x ( Ω Θ ω x p d μ ( ω ) ) 1 p , Open image in new window
therefore
( 1 λ 1 2 λ 2 ) A γ 1 λ 2 x ( Ω Θ ω x p d μ ( ω ) ) 1 p . Open image in new window

 □

Acknowledgements

The authors would like to sincerely thank Prof. Dr. Gitta Kutyniok and Dr. Asghar Ranjbari for their valuable comments.

Copyright information

© Faroughi and Rahmani; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesUniversity of TabrizTabrizIran
  2. 2.Islamic Azad University-Shabestar BranchShabestarIran