Abstract. Casazza and Christensen in [5], introduced and studied the reconstruction property in Banach spaces. In this paper sufficient conditions for the ...
Palestine Journal of Mathematics Vol. 3(1) (2014) , 11–26
© Palestine Polytechnic University-PPU 2014
Reconstruction Property and Frames in Banach Spaces S. K. Kaushik, L. K. Vashisht and G. Khattar Communicated by Akram Aldroubi MSC 2010 Classifications: 42C15; 42C30; 42C05; 46B15. Keywords and phrases: Frames, Banach frame, Perturbation, Besselian frames, Reconstruction property. The authors thank the referee(s) for giving constructive comments and suggestions towards the improvement of the paper.
Abstract. Casazza and Christensen in [5], introduced and studied the reconstruction property in Banach spaces. In this paper sufficient conditions for the existence of the reconstruction property in Banach spaces are obtained. Casazza and Christensen gave Paley- Wiener type perturbation of the reconstruction property which does not force equivalence of the sequences. Some Paley-Wiener type perturbations concerning the reconstruction property are discussed. Motivated by a paper by Holub [18], the notion of Besselian type reconstruction property in Banach spaces is introduced and its application to Banach frames is obtained.
1 Introduction The Fourier transform has been widely used in analysis for more than a century. However, it only provides frequency information, and hides (in its phases) information concerning the moment of emission and duration of a signal. D. Gabor in 1946, introduced a fundamental approach to signal decomposition in terms of elementary signals and resolve this problem [14]. Duffin and Schaeffer [11] in 1952, while addressing some deep problems in non-harmonic Fourier series, abstracted Gabor’s method to define frames for Hilbert spaces. Later, in 1986, Daubechies, Grossmann and Meyer [10] found new applications to wavelet and Gabor transforms in which frames played an important role. Let H be a separable Hilbert space. A countable system {fn } ⊂ H is called frame (Hilbert) for H if there exists positive constants A and B such that Akf k2 ≤ k{hf, fn i}k2`2 ≤ Bkf k2 ,
for all f ∈ H.
The positive constants A and B , respectively, are called lower and upper frame bounds of the are not unique. The operator T : `2 → H defined as T ({ck }) = P∞frame {fn }. They 2 k=1 ck fk , {ck } ∈ ` , is called the pre-frame operator or the synthesis operator and its adjoint T ∗ : H → `2 given by T ∗ (f ) = {hf, fk i}, for all f ∈ H, is called the analysis operator. Composing T and T ∗ we obtain the frame operator S = T T ∗ : H → H given by S (f ) =
∞ X hf, fk ifk , for all f ∈ H. k=1
The frame operator S is a positive, self-adjoint invertible operator on H. For all f ∈ H, we have f = SS −1 f =
∞ ∞ X X hS −1 f, fk ifk = hf, S −1 fk ifk . k=1
k=1
The series converges unconditionally and is called the reconstruction formula for the frame. The representation of f in the reconstruction formula need not be unique. Today, frames play
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S. K. Kaushik, L. K. Vashisht and G. Khattar
important roles in many applications in mathematics, science and engineering. In the theoretical direction, powerful tools from operator theory and Banach spaces are being employed to study frames. For a nice introduction to theory of frames an interested reader may refer to [1, 2, 8, 17, 34] and references therein. During the development of redundant building blocks (elementary signals), in the later half of twentieth century, Coifman and Weiss in [9] introduced the notion of atomic decomposition for function spaces. Later, Feichtinger and Gröchenig [12, 13] extended this idea to Banach spaces. This concept was further generalized by Gröchenig [15] who introduced the notion of Banach frames for Banach spaces. Casazza, Han and Larson [4] also carried out a study of atomic decompositions and Banach frames. Recently, various generalization of frames in Banach spaces have been introduced and studied. Han and Larson [16] defined a Schauder frame for a Banach space X to be an inner direct summand (i.e. a compression) of a Schauder basis of X . Schauder frames were further studied in [22, 25, 26, 31]. The reconstruction property in Banach spaces was introduced and studied by Casazza and Christensen in [5] and further studied in [32]. The reconstruction property is an important tool in several areas of mathematics and engineering. As the perturbation result of Paley and Wiener preserves reconstruction property, it becomes more important from an application point of view. Further, the reconstruction property is used as a tool to recover certain Banach spaces. The reconstruction property is also used to study the geometry of Banach spaces. In fact, it is related to bounded the approximated property as observed in [1, 3, 4]. In [5], Casazza and Christensen gave some perturbation results. In fact, they develop a more general perturbation theory that does not force equivalence of the frames. This paper is organized as follows: In Section 2 we give basic definitions and results which will be used throughout the paper. Sufficient conditions for the existence of the reconstruction property are discussed in Section 3. Section 4 is devoted to perturbation of reconstruction property. Casazza and Christensen give Paley- Wiener type perturbation of reconstruction property which does not force equivalence of the sequences. A perturbation result concerning the reconstruction property in which equivalence of sequences is one of the sufficient conditions is given. Uniform approximation of a compact operator on a Banach space which admits a reconstruction property is discussed. By inspiration from a paper by Holub [18], we introduce the notion of Besselian type reconstruction property in Banach spaces in Section 5. An application of the Besselian type reconstruction property to Banach frames have also been obtained. Banach frames for operator spaces associated with the reconstruction property are discussed in Section 6.
2 Preliminaries Throughout this paper X will denote an infinite dimensional Banach space over the scalar field K (which will be R or C), X ∗ the conjugate space (topological) of X . The map π : X → X ∗∗ denotes the canonical mapping from X into X ∗∗ . The closure of the linear hull of a system {fn } ⊂ X in the norm topology of X is denoted by [fn ]. The space of all bounded linear operators from a Banach space X into a Banach space Y is denoted by B(X , Y ). For a pair (P {fk }, {fk∗ }) ⊂ X × X ∗ , {Pn } is the sequence of finite rank operators defined by Pn (f ) = n ∗ 2 k=1 fk (f )fk , f ∈ X . The sequence of canonical unit vectors in ` is denoted by {ek }. Definition 2.1. [15] Let X be a Banach space and let Xd be an associated Banach space of scalar valued sequences indexed by N. Let {fk∗ } ⊂ X ∗ and S : Xd → X be given. The pair ({fk∗ }, S ) is called a Banach frame for X with respect to Xd if (i) {fk∗ (f )} ∈ Xd , for all f ∈ X . (ii) There exist positive constants A and B with 0 < A ≤ B < ∞ such that Akf kX ≤ k{fk∗ (f )}kXd ≤ Bkf kX ,
for all f ∈ X .
(2.1)
Reconstruction Property and Frames in Banach Spaces
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(iii) S is a bounded linear operator such that S ({fk∗ (f )}) = f,
for all f ∈ X .
Definition 2.2. [5] Let X be a separable Banach space . A sequence {fk∗ } ⊂ X ∗ has the reconstruction property for X with respect to a sequence {fk } ⊂ X if f=
∞ X
fn∗ (f )fn ,
for all f ∈ X .
(2.2)
n=1
In short, we will say that the pair ({fk }, {fk∗ }) has the reconstruction property for X . More precisely, we say that ({fk }, {fk∗ }) is a reconstruction system or reconstruction property for X . Remark 2.3. An interesting example for the reconstruction property is given in [5]: Let {fk∗ } ⊂ `∞ and {fk∗ } is unitarily equivalent to the unit vector basis of `2 . Then, {fk∗ } has the reconstruction property with respect to its own pre-dual (that is, expansions with respect to the orthonormal basis). Regarding the existence of Banach spaces which have a reconstruction system, Casazza and Christensen proved the following result. Proposition 2.4. [5] There exists a Banach space X with the following properties: P ∗ (i) There is a sequence {fk } such that each f ∈ X has a expansion f = ∞ k=1 fk (f )fk . (ii) X does not have the reconstruction property with respect to any pair ({hk }, {h∗k }). Definition 2.5. [4] A separable Banach space X has the λ-bounded approximation property (i.e. λ-BAP) if there is a sequence of finite rank operators {Ti } defined on X so that for every f ∈ X , Ti f → f in norm. We say that X has the Bounded approximation property (denoted by BAP) if X has the λ-BAP, for some λ. The notion of reconstruction property is related to the Bounded Approximation Property(BAP). If ({fk }, {fk∗ }) has the reconstruction property for X , then X has the bounded approximation property. Conversely, if X has the bounded approximation property then there exists a Banach space A ⊃ X with a basis and by using a projection P : A → X we can find a sequence {gk∗ } ⊂ X ∗ such that {gk∗ } has reconstruction property for X with respect to {P (•)}k . So, X is isomorphic to a complemented subspace of a Banach space with a basis. The reconstruction property is also used to study geometry of Banach spaces [3]. For more results and basics on the reconstruction property and bounded approximation property one may refer to [4] and references therein.
3 Reconstruction Property in Banach Spaces Suppose that each vector of a Banach space X is expressed as an infinite linear combination of a given system say {fk } ⊂ X . Then, a natural question arises to find further condition(s) which guarantee the existence of {fk∗ } ⊂ X ∗ such that ({fk }, {fk∗ }) has the reconstruction property for X . This problem is very deep and we do not know the answer even for Hilbert spaces. In this direction the following proposition gives sufficient conditions for the existence of a sequence {fk∗ } ⊂ X ∗ such that ({fk }, {fk∗ }) has the reconstruction property for X . Proposition 3.1. Let {fk }\{0} ⊂ X be a sequence of vectors such that for each f ∈ X , there ex∞ P P∞ ists a sequence {γk } ⊂ K such that f = γk fk . Let Y = {{γk } ⊂ K : k=1 γk fk converges in the norm in X } k=1
n
P
P∞
be a Banach space with norm given by k{γk }kY = sup γk fk
. If Z = {{γk } ⊂ K : k=1 γk fk = 0} 1≤n
