Journal of Inequalities and Applications

, 2012:196

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
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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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq1_HTML.gif-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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif is a family of Hilbert spaces.

Suppose that ( Ω , Σ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq4_HTML.gif is called Bochner measurable if there exists a sequence of simple functions { f n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq5_HTML.gif such that
lim n f n ( ω ) f ( ω ) = 0 , a.e.  [ μ ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equa_HTML.gif
Definition 1.2 If μ is a measure on ( Ω , Σ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq6_HTML.gif then X has the Radon-Nikodym property with respect to μ if for every countably additive vector measure γ on ( Ω , Σ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq6_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq7_HTML.gif such that
γ ( E ) = E g ( ω ) d μ ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equb_HTML.gif

for every set E Σ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq8_HTML.gif.

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gif is a measure space and X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq9_HTML.gif has the Radon-Nikodym property. Let 1 p https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq10_HTML.gif. The Bochner space of L p ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq11_HTML.gif 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 = https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equc_HTML.gif
are finite. In [10, 11] and [12], p.51] it is proved that if 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq12_HTML.gif and q is such that 1 p + 1 q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq13_HTML.gif, then L q ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq14_HTML.gif is isometrically isomorphic to ( L p ( μ , X ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq15_HTML.gif if and only if X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq9_HTML.gif has the Radon-Nikodym property. This isometric isomorphism is the mapping
ψ : L q ( μ , X ) ( L p ( μ , X ) ) , g ψ ( g ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equd_HTML.gif
where the mapping ψ ( g ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq16_HTML.gif is defined on L p ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq11_HTML.gif by
ψ ( g ) ( f ) = Ω g ( ω ) ( f ( ω ) ) d μ ( ω ) , f L p ( μ , X ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Eque_HTML.gif
So for all f L p ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq17_HTML.gif and g L q ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq18_HTML.gif, we have
f , ψ ( g ) = Ω f ( ω ) , g ( ω ) d μ ( ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equf_HTML.gif
In the following, we use the notation f , g https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq19_HTML.gif instead of f , ψ ( g ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq20_HTML.gif, so for all f L p ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq17_HTML.gif and g L q ( μ , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq18_HTML.gif,
f , g = Ω f ( ω ) , g ( ω ) d μ ( ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equg_HTML.gif
Particularly, if H is a Hilbert space, then ( L p ( μ , H ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq21_HTML.gif is isometrically isomorphic to L q ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq22_HTML.gif. So, for all f L p ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq23_HTML.gif and g L q ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq24_HTML.gif,
f , g = Ω f ( ω ) , g ( ω ) d μ ( ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equh_HTML.gif
in which f ( ω ) , g ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq25_HTML.gif does not mean the inner product of elements f ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq26_HTML.gif, g ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq27_HTML.gif in H, but
f ( ω ) , g ( ω ) = ν ( g ( ω ) ) ( f ( ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equi_HTML.gif

where ν : H H https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq28_HTML.gif is the isometric isomorphism between H and H https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq29_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq30_HTML.gifis a bounded operator from a Banach spaceXinto a Banach spaceY, then its adjoint U : Y X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq31_HTML.gifis surjective if and only ifUhas a bounded inverse on R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq32_HTML.gif.

Note that for a collection { H β } β B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq33_HTML.gif of Hilbert spaces, we can suppose that there exists a Hilbert space K such that for all β B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq34_HTML.gif, H β K https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq35_HTML.gif, where K = β B H β https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq36_HTML.gif is the direct sum of { H β } β B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq37_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq38_HTML.gif is a scalar function of time and space, one can write ( f ( t ) ) ( x ) : = g ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq39_HTML.gif to make f a function of time, with f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq40_HTML.gif 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 < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq41_HTML.gif. The family { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq42_HTML.gif is a Bochner pg-frame for X with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq43_HTML.gif if:
  1. (i)

    For each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq45_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ1_HTML.gif
    (2.1)
     

A and B are called the lower and upper Bochner pg-frame bounds respectively. We call that { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a tight Bochner pg-frame if A and B can be chosen such that A = B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq47_HTML.gif, and a Parseval Bochner pg-frame if A and B can be chosen such that A = B = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq48_HTML.gif. If for each ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq49_HTML.gif, H ω = H https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq50_HTML.gif, then { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq51_HTML.gif is called a Bochner pg-frame for X with respect to H. A family { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq52_HTML.gif is called a Bochner pg-Bessel family for X with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq53_HTML.gif be a frame for Hilbert space H, Ω = I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq54_HTML.gif and μ be a counting measure on Ω. Set
Λ i : H C , Λ i ( h ) = h , f i , h H . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equj_HTML.gif

Then { Λ i } i I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq55_HTML.gif is a Bochner pg-frame for H with respect to C https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq56_HTML.gif.

Example 2.3 Let Ω = { a , b , c } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq57_HTML.gif, Σ = { , { a , b } , { c } , Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq58_HTML.gif and μ : Σ [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq59_HTML.gif be a measure such that μ ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq60_HTML.gif, μ ( { a , b } ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq61_HTML.gif, μ ( { c } ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq62_HTML.gif and μ ( Ω ) = 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq63_HTML.gif. Assume that X = L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq64_HTML.gif and { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif is a family of arbitrary Hilbert spaces and consider a fixed family { h ω } ω Ω { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq65_HTML.gif such that h ω = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq66_HTML.gif, ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq49_HTML.gif. Suppose that
Λ ω : L p ( Ω ) H ω , Λ ω ( φ ) = φ ( c ) h ω . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equk_HTML.gif
It is clear that Λ ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq67_HTML.gif’s are bounded and for each φ L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq68_HTML.gif, ω Λ ω ( φ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq69_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equl_HTML.gif

So, { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a Bochner pg-frame for L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq70_HTML.gif with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif.

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

Definition 2.4 Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif be a Bochner pg-Bessel family for X with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif, and q be the conjugate exponent of p. We define the operators T and U, by
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ2_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ3_HTML.gif
(2.3)

The operators T and U are called the synthesis and analysis operators of { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif 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 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq52_HTML.gifbe a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifand with Bessel boundB. Then the operatorsTandUdefined by (2.2) and (2.3) respectively, are well defined and bounded with T B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq71_HTML.gifand U B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq72_HTML.gif.

Proof Suppose that { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a Bochner pg-Bessel family with bound B, and q is the conjugate exponent of p. We show that for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif and all G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif, the mapping ω Λ ω ( x ) , G ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq74_HTML.gif is measurable. For all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif and G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq75_HTML.gif and G are Bochner measurable, so there are sequences of simple functions { λ n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq76_HTML.gif and { g n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq77_HTML.gif 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equm_HTML.gif
For each n, λ n , g n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq78_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equn_HTML.gif
So
lim n | Λ ω ( x ) , G ( ω ) λ n ( ω ) , g n ( ω ) | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equo_HTML.gif

and ω Λ ω ( x ) , G ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq79_HTML.gif is measurable.

For each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif and G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq80_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equp_HTML.gif

Thus T is well defined, and T B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq71_HTML.gif. By a similar discussion, U is well defined and U B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq72_HTML.gif. □

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

Proposition 2.6If { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifthen for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ( U x ) ( ω ) = Λ ω x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq81_HTML.gif, a.e. [ μ ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq82_HTML.gif.

Proof Let q be the conjugate exponent of p and x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif. For all G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif, we have
U x , G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) = { Λ ω x } ω Ω , G . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equq_HTML.gif
So U x { Λ ω x } ω Ω , G = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq83_HTML.gif, for all G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif. There exists G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif such that G q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq84_HTML.gif and
U x { Λ ω x } ω Ω , G = U x { Λ ω x } ω Ω p , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equr_HTML.gif

which implies U x { Λ ω x } ω Ω p = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq85_HTML.gif. Therefore, ( U x ) ( ω ) = Λ ω x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq81_HTML.gif, a.e. [ μ ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq82_HTML.gif. □

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifis a measure space whereμisσ-finite. Let { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq86_HTML.gifbe a family such that for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq87_HTML.gifis Bochner measurable, and assume that there exist positive constantsAandBsuch that (2.1) holds for allxin a dense subsetVofX. Then { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq88_HTML.gifis a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifwith boundsAandB.

Proof Let { Ω n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq89_HTML.gif be a family of disjoint measurable subsets of Ω such that Ω = n = 1 Ω n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq90_HTML.gif with μ ( Ω n ) < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq91_HTML.gif for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq92_HTML.gif. Let x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif and assume, without loss of generality, Λ ω x 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq93_HTML.gif, ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq94_HTML.gif. Let
Δ m x = { ω Ω | m 1 < Λ ω x m } , m = 0 , 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equs_HTML.gif
It is clear that for each m = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq95_HTML.gif , Δ m x Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq96_HTML.gif is measurable and Ω = m = 0 , n = 1 ( Δ m x Ω n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq97_HTML.gif, where { Δ m x Ω n } n = 1 , m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq98_HTML.gif is a family of disjoint and measurable subsets of Ω. If { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is not a Bochner pg-Bessel family for X, then there exists x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif such that
( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p > B x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equt_HTML.gif
So
m , n Δ m x Ω n Λ ω ( x ) p d μ ( ω ) > B p x p https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equu_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ4_HTML.gif
(2.4)
Let { x k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq99_HTML.gif be a sequence in V such that x k x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq100_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq101_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equv_HTML.gif
which is a contradiction to (2.4) (by the Lebesgue’s Dominated Convergence Theorem). So { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a pg-Bessel family for X with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq102_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equw_HTML.gif
Since { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equx_HTML.gif
and
G k : Ω ω Ω H ω , G k ( ω ) = Λ ω x k p q q Λ ω x k , ω Ω . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equy_HTML.gif
It is obvious that for each k N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq103_HTML.gif, G k https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq104_HTML.gif and G belong to L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq105_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equz_HTML.gif
Since lim k ( G k G ) ( ω ) q = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq106_HTML.gif and
( G k G ) ( ω ) q ( G k ( ω ) + G ( ω ) ) q 2 q 1 ( G k ( ω ) q + G ( ω ) q ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaa_HTML.gif
so by the Lebesgue’s Dominated Convergence Theorem,
lim k Ω ( G k G ) ( ω ) q d μ ( ω ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equab_HTML.gif
Therefore, lim k G k G q = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq107_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equac_HTML.gif

By letting x k x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq100_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq108_HTML.gif-spaces.

Lemma 2.8Let ( Ω , Σ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifbe a measure space whereμisσ-finite. Let 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq109_HTML.gifandqbe its conjugate exponent. If F : Ω H https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq110_HTML.gifis Bochner measurable and for each G L q ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq111_HTML.gif, | Ω F ( ω ) , G ( ω ) H d μ ( ω ) | < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq112_HTML.gif, then F L p ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq113_HTML.gif.

Proof Let { Ω n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq89_HTML.gif be a family of disjoint measurable subsets of Ω such that for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq92_HTML.gif, μ ( Ω n ) < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq114_HTML.gif and Ω = n = 1 Ω n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq90_HTML.gif. Without loss of generality, we can assume F ( ω ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq115_HTML.gif, ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq49_HTML.gif. Let
Δ m = { ω Ω | m 1 < F ( ω ) m } , m = 0 , 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equad_HTML.gif
It is clear that for each m = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq95_HTML.gif , Δ m Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq116_HTML.gif is measurable and Ω = m = 0 , n = 1 ( Δ m Ω n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq117_HTML.gif, where { Δ m Ω n } n = 1 , m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq118_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equae_HTML.gif
and
Δ m Ω n F ( ω ) p d μ ( ω ) m p μ ( Ω n ) < . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaf_HTML.gif
Suppose that Ω F ( ω ) p d μ ( ω ) = https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq119_HTML.gif, then there exists a family { E k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq120_HTML.gif of disjoint finite subsets of N 0 × N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq121_HTML.gif such that
( m , n ) E k Δ m Ω n F ( ω ) p d μ ( ω ) > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equag_HTML.gif
Let E = k = 1 ( m , n ) E k ( Δ m Ω n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq122_HTML.gif. Consider G : Ω H https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq123_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equah_HTML.gif
where
c k : = 1 k q p ( ( m , n ) E k ( Δ m Ω n ) F ( ω ) p d μ ( ω ) ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equai_HTML.gif
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 < . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaj_HTML.gif
Therefore, G L q ( μ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq111_HTML.gif. 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 = , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equak_HTML.gif

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifis a measure space whereμisσ-finite. Let { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq86_HTML.gifbe a family such that for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gifthe mapping ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq75_HTML.gifis Bochner measurable. If the operatorTdefined by (2.2) is well defined and bounded, then { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifwith Bessel bound T https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq124_HTML.gif.

Proof Let q be the conjugate exponent of p and for x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equal_HTML.gif
Then F x ( L q ( μ , ω Ω H ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq125_HTML.gif. So { Λ ω x } ω Ω L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq126_HTML.gif by Lemma 2.8. By Remark 1.3, ( L q ( μ , ω Ω H ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq127_HTML.gif and L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq128_HTML.gif are isometrically isomorphic and { Λ ω x } ω Ω p = F x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq129_HTML.gif. Therefore,
( Ω Λ ω x p d μ ( ω ) ) 1 p = F x = sup G q = 1 | x , T G | T x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equam_HTML.gif

 □

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

Lemma 2.10Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equan_HTML.gif
By Proposition 2.6, we have
A x U x p B x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equao_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif. ThenXis reflexive.

Proof By Lemma 2.10, R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq32_HTML.gif is a closed subspace of L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq128_HTML.gif and U : X R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq130_HTML.gif is homeomorphism. Since L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq128_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-Bessel family forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq102_HTML.gifwith the synthesis operatorTand the analysis operatorU. Then

(i) U = T https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq131_HTML.gif.

(ii) If { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifhas the lower Bochnerpg-frame condition, then T J 1 = ψ J 2 U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq132_HTML.gif, where
J 1 : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equap_HTML.gif
and
J 2 : L p ( μ , ω Ω H ω ) ( L p ( μ , ω Ω H ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaq_HTML.gif

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

Proof (i) For each G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gif and x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, we have
U x , G = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) = x , T G , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equar_HTML.gif
so U = T https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq131_HTML.gif.
  1. (ii)
    Since X and L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq133_HTML.gif are reflexive, J 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq134_HTML.gif and J 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq135_HTML.gif are surjective. For each G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq80_HTML.gif and x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif,
    G , T J 1 x = T G , J 1 x = x , T G = U x , G , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equas_HTML.gif
     
also
G , ψ J 2 U x = ψ G , J 2 U x = U x , ψ G = U x , G . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equat_HTML.gif

Hence T J 1 = ψ J 2 U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq132_HTML.gif. □

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 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq86_HTML.gif.

(i) Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif. Then the operatorTdefined by (2.2) is a surjective bounded operator.

(ii) Let ( Ω , Σ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifbe a measure space whereμisσ-finite and for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq45_HTML.gifbe Bochner measurable. Let the operatorTdefined by (2.2) be a surjective bounded operator. Then { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif.

Proof (i) Since { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq131_HTML.gif is surjective.
  1. (ii)
    Since T is bounded, { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq136_HTML.gif is a Bochner pg-Bessel family, by Theorem 2.9. Since T = U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq137_HTML.gif is surjective, U has a bounded inverse on R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq32_HTML.gif by Lemma 1.4. So there exists A > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq138_HTML.gif such that for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, U x p A x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq139_HTML.gif. By Proposition 2.6, for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif
    A x U x p = ( Ω Λ ω ( x ) p d μ ( ω ) ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equau_HTML.gif
     

Hence { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a Bochner pg-frame. □

Corollary 2.14If { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifandqis the conjugate exponent ofp, then for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq140_HTML.gif, there exists G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gifsuch that
x , x = Ω Λ ω ( x ) , G ( ω ) d μ ( ω ) , x X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equav_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif. Then T https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq141_HTML.gifand U ˜ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq142_HTML.gifare the optimal upper and lower Bochnerpg-frame bounds of { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifrespectively, where U ˜ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq143_HTML.gifis the inverse ofUon R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq32_HTML.gif, andT, Uare the synthesis and analysis operators of { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq144_HTML.gifrespectively.

Proof From the proof of Theorem 2.9, for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, we have
( Ω Λ ω x p d μ ( ω ) ) 1 p = F x = sup G q = 1 | x , T G | . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaw_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equax_HTML.gif
By Proposition 2.6, U x p = ( Ω Λ ω x p d μ ( ω ) ) 1 p https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq145_HTML.gif; consequently,
inf x = 1 U x p = inf x = 1 ( Ω Λ ω x p d μ ( ω ) ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equay_HTML.gif
The operator U : X L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq146_HTML.gif is bounded below, so it has bounded inverse U ˜ : R U X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq147_HTML.gif. 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 ˜ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equaz_HTML.gif

hence inf x = 1 ( Ω Λ ω x p d μ ( ω ) ) 1 p = 1 U ˜ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq148_HTML.gif. □

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 < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq149_HTML.gif. A family { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq42_HTML.gif is called a Bochner qg-Riesz basis for X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gif with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif, if:
  1. (i)

    { x : Λ ω x = 0 ,  a.e.  [ μ ] } = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq151_HTML.gif,

     
  2. (ii)
    for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq45_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equba_HTML.gif
     

A and B are called the lower and upper Bochner qg-Riesz basis bounds of { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif respectively.

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

Proposition 3.2Suppose that ( Ω , Σ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifis a measure space whereμisσ-finite and consider the family { Λ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq52_HTML.gif.

(i) Assume that for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Λ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq152_HTML.gifis Bochner measurable. { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerqg-Riesz basis for X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gifwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq153_HTML.gifif and only if the operatorTdefined by (2.2) is an invertible bounded operator from L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq154_HTML.gifonto X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gif.

(ii) Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerqg-Riesz basis for X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gifwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq102_HTML.gifwith the optimal upper Bochnerqg-Riesz basis boundB. Ifpis the conjugate exponent ofq, then { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifwith 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq51_HTML.gif is a Bochner pg-frame for X with respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifis a measure space whereμisσ-finite. Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifwith the synthesis operatorTand the analysis operatorU, andqbe the conjugate exponent ofp. Then the following statements are equivalent:

(i) { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifis a Bochnerqg-Riesz basis for X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gif.

(ii) Tis injective.

(iii) R U = L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq155_HTML.gif.

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq156_HTML.gif, and hence { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif is a Bochner qg-Riesz basis for X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq150_HTML.gif.

     
  2. (i)

    → (iii): By Theorem 3.2, T is invertible, so T https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq157_HTML.gif is invertible. Lemma 2.12(ii) implies that R U = L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq158_HTML.gif.

     
  3. (iii)

    → (i): Since the operator U is invertible, by Lemma 2.12, T = U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq137_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq3_HTML.gifis a measure space whereμisσ-finite. Let { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gifbe a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifandqbe the conjugate exponent of p. Let { Θ ω B ( X , H ω ) : ω Ω } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq159_HTML.gifbe a family such that for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, ω Θ ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq160_HTML.gifis Bochner measurable. Assume that there exist constants λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq161_HTML.gif, λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq162_HTML.gif, γsuch that
0 λ 2 < 1 , λ 2 λ 1 < 1 , 0 γ < ( 1 λ 1 2 λ 2 ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbb_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ5_HTML.gif
(4.1)
for all G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq73_HTML.gifand x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif. Then { Θ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq163_HTML.gifis a Bochnerpg-frame forXwith respect to { H ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq2_HTML.gifwith bounds
A [ ( 1 λ 1 2 λ 2 ) γ A 1 λ 2 ] and B [ 1 + λ 1 + γ B 1 λ 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbc_HTML.gif

whereAandBare the Bochnerpg-frame bounds for { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq46_HTML.gif.

Proof For each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif and G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq80_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbd_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Eqube_HTML.gif
Now, define W : L q ( μ , ω Ω H ω ) X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq164_HTML.gif by
x , W G = Ω Θ ω x , G ( ω ) d μ ( ω ) , x X , G L q ( μ , ω Ω H ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbf_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbg_HTML.gif

so W is well defined and bounded. By Theorem 2.9, { Θ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq163_HTML.gif is a Bochner pg-Bessel family for X with bound B [ 1 + λ 1 + γ B 1 λ 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq165_HTML.gif.

Now, we show that { Θ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq163_HTML.gif satisfies the lower Bochner pg-frame condition. Let T and U be the synthesis and analysis operators of { Λ ω } ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq88_HTML.gif respectively. By Proposition 2.6, for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif,
A x U x p B x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbh_HTML.gif
By Lemma 2.10, R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq32_HTML.gif is a closed subspace of L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq128_HTML.gif, so Q = U : X R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq166_HTML.gif is a bijective bounded operator, hence ( Q 1 ) : X R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq167_HTML.gif is alike. Since B 1 Q 1 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq168_HTML.gif, so ( Q 1 ) = Q 1 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq169_HTML.gif. Let x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq140_HTML.gif and S = ( Q 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq170_HTML.gif, then S A 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq171_HTML.gif and S ( x ) R U https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq172_HTML.gif, by Hahn-Banach theorem there exists φ L p ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq173_HTML.gif such that φ | R U = S ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq174_HTML.gif and φ = S ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq175_HTML.gif. It follows that
φ = S ( x ) S x A 1 x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ6_HTML.gif
(4.2)
By Remark 1.3, there exists G L q ( μ , ω Ω H ω ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq176_HTML.gif such that ψ ( G ) = φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq177_HTML.gif, then
G q = φ A 1 x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equ7_HTML.gif
(4.3)
Since x = Q ( Q 1 ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq178_HTML.gif, we have for each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif,
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbi_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbj_HTML.gif
which implies
W G x ( λ 1 + λ 2 ) A + γ ( 1 λ 2 ) A x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbk_HTML.gif
For a given x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq44_HTML.gif, there exists x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_IEq140_HTML.gif such that
x = 1 , x = x ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbl_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbm_HTML.gif
therefore
( 1 λ 1 2 λ 2 ) A γ 1 λ 2 x ( Ω Θ ω x p d μ ( ω ) ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-196/MediaObjects/13660_2012_Article_306_Equbn_HTML.gif

 □

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