# Approximative $K$-Atomic Decompositions and frames in Banach Spaces

Shah Jahan  Shah Jahan, Department of Mathematics, University of Delhi,
Delhi-110007, India<EMAIL_ADDRESS>
###### Abstract.

[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012),
139-144] introduced a special kind of frames, named $K$-frames, where $K$ is
an operator, in Hilbert spaces, is significant in frame theory and has many
applications. In this paper, first of all, we have introduced the notion of
approximative $K$-atomic decomposition in Banach spaces. We gave two
characterizations regarding the existence of approximative $K$-atomic
decompositions in Banach spaces. Also some results on the existence of
approximative $K$-atomic decompositions are obtained. We discuss several
methods to construct approximative $K$-atomic decomposition for Banach Spaces.
Further, approximative $\mathcal{X}_{d}$-frame and approximative
$\mathcal{X}_{d}$-Bessel sequence are introduced and studied. Two necessary
conditions are given under which an approximative $\mathcal{X}_{d}$-Bessel
sequence and approximative $\mathcal{X}_{d}$-frame give rise to a bounded
operator with respect to which there is an approximative $K$-atomic
decomposition. Examples and counter examples are provided to support our
concept. Finally, a possible application is given.

###### Key words and phrases:

Frames; $K$-frames; Atomic decomposition; K-Atomic decomposition;
$\mathcal{X}_{d}$-Bessel sequence ; $\mathcal{X}_{d}$-frames

###### 2010 Mathematics Subject Classification:

42A38, 42C15, 42C30, 46B15

##  1\. Introduction and Preliminaries

Fourier transform has been a major tool in analysis for over a century. It has
a serious lacking for signal analysis in which it hides in its phase
information concerning the moment of emission and duration of a signal. What
actually needed was a localized time frequency representation which has this
information encoded in it. In 1946, Dennis Gabor [13] filled this gap and
formulated a fundamental approach to signal decomposition in terms of
elementary signals. On the basis of this development, in 1952, Duffin and
Schaeffer [9] introduced frames for Hilbert spaces to study some deep problems
in non-harmonic Fourier series. In fact, they abstracted the fundamental
notion of Gabor for studying signal processing. Let $\mathcal{H}$ be a real
(or complex) separable Hilbert space with inner product $\langle.,.\rangle.$ A
countable sequence $\\{f_{n}\\}\subset\mathcal{H}$ is called a frame for the
Hilbert space $\mathcal{H},$ if there exist positive constants $A,~{}B>0$ such
that

$\displaystyle A\|f\|^{2}_{\mathcal{H}}\leq\sum_{n=1}^{\infty}|\langle
f,f_{n}\rangle|^{2}\leq B\|f\|^{2}_{\mathcal{H}},~{}~{}~{}\mbox{for
all}~{}~{}\in\mathcal{H}$ (1.1)

The positive numbers $A$ and $B$ are called the lower and upper frame bounds
of the frame, respectively. These bounds are not unique. The inequality in
(1.1) is called the frame inequality of the frame. If $\\{f_{n}\\}$ is a frame
for $\mathcal{H}$ then the following operators are associated with it.

  1. (a)

Pre-frame operator $T:l^{2}(\mathbb{N})\longrightarrow\mathcal{H}$ is defined
as
$T\\{c_{n}\\}_{n=1}^{\infty}=\sum\limits_{k=1}^{\infty}c_{n}f_{n},~{}~{}\\{c_{n}\\}_{n=1}^{\infty}\in
l^{2}(\mathbb{N}).$

  2. (b)

Analysis operator $T^{*}:\mathcal{H}\longrightarrow
l^{2}(\mathbb{N}),T^{*}=\\{\langle
f,f_{k}\rangle\\}_{k=1}^{\infty}~{}~{}~{}f\in\mathcal{H}.$

  3. (c)

Frame operator
$S=TT^{*}=:\mathcal{H}\longrightarrow\mathcal{H},~{}~{}S_{f}=\sum\limits_{k=1}^{\infty}\langle
f,f_{k}\rangle f_{k},~{}~{}f\in\mathcal{H}.$ The frame $S$ is bounded, linear
and invertible on $\mathcal{H}$. Thus, a frame for $\mathcal{H}$ allows each
vector in $\mathcal{H}$ to be written as a linear combination of the elements
in the frame, but the linear independence between the elements is not
required; i.e for each vector $f\in\mathcal{H}$ we have,

$\displaystyle f=SS^{-1}f=\sum\limits_{k=1}^{\infty}\langle f,f_{k}\rangle
f_{k}.$

For more details related to frame and Riesz bases in Hilbert spaces, one may
refer to [4, 7]. These ideas did not generate much interest outside of non-
harmonic Fourier series and signal processing for more than three decades
until Daubechies et al. [8] reintroduced frames. After this landmark paper the
theory of frames begin to be studied widely and found many applications to
wavelet and Gabor transforms in which frames played an important role.
Feichtinger and Gröcheing [11] extended the idea of Hilbert frames to Banach
spaces and called it atomic decomposition. A more general concept called
Banach frame was introduced by Gröcheing [15] and were further studied in [19,
31]. Banach frames were developed for the theory of frames in the context of
Gabor and Wavelet analysis. Christensen and Heil [6] studied some perturbation
results for Banach frames and atomic decompositions.
In particular, frames are widely used in sampling theory in [1] amounts to the
construction of Banach frames consisting of reproducing kernals for a large
class of shift invariant spaces. Aldroubi et al. [2] used Banach frames in
various irregular sampling problems. Elder and Forney [10] used tight frames
for quantum measurement. Gröchenig [16] emphasised that localization of a
frame is a necessary condition for its extension to a Banach frame for the
associated Banach spaces. He also observed that localized frames are universal
Banach frames for the associated family of Banach spaces. Fornasier [12]
studied Banach frames for $\alpha$-modulation spaces. In fact, he gave a
Banach frame characterization for the $\alpha$-modulation spaces. Shah et.al
[18] defined and studied Banach frames to a new geometric notation; in fact
they gave a sufficient condition and a necessary condition for a cone
associated with a Banach frame to be a generating cone.
Casazza et al. [5] studied $\mathcal{X}_{d}$-frames and
$\mathcal{X}_{d}$-Bessel sequences in Banach spaces. Stoeva [27] gave some
perturbation results for $\mathcal{X}_{d}$-frames and atomic decompositions.
Kaushik and Sharma [20] studied approximative atomic decompositions in Banach
spaces. For further studies related to approximative frame one may refer [17,
21, 25]. Gavruta [14] introduced and studied atomic system for an operator $K$
and the notion of $K$-frame in a Hilbert space. Xiao et al. [30] discussed
relationship between $K$-frames and ordinary frames in Hilbert spaces. Poumai
and Jahan [23] introduced K-atomic decompositions in Banach spaces.
outline of the paper. In this paper, we have introduced the notion of
approximative $K$-atomic decomposition in Banach spaces. We gave two
characterizations regarding the existence of approximative $K$-atomic
decompositions in Banach spaces. Also some results on the existence of
approximative $K$-atomic decompositions are obtained. We discuss several
methods to construct approximative $K$-atomic decomposition for Banach Spaces.
Further, approximative $\mathcal{X}_{d}$-frame and approximative
$\mathcal{X}_{d}$-Bessel sequence are introduced and studied. Two necessary
conditions are given under which an approximative $\mathcal{X}_{d}$-Bessel
sequence and approximative $\mathcal{X}_{d}$-frame give rise to bounded
operators with respect to which there is an approximative $K$-atomic
decomposition. Examples and counter examples are provided to support our
concept of approximative K-atomic decomposition. Finally, we gave a possible
application of our work.
Next we give some basic notations. Throughout this paper, $\mathcal{X}$ will
denote a separable Banach space over the scalar field K($\mathbb{R}$ or
$\mathbb{C}$), $\mathcal{X}^{*}$ the dual space of $\mathcal{X}$,
$\mathcal{X}_{d}$ a BK-space and $L(\mathcal{X},\mathcal{Y})$ will denote the
space of all bounded linear operators from $\mathcal{X}$ into $\mathcal{Y}$.
For $T\in L(\mathcal{X})$, $T^{*}$ denotes the adjoint of $T$,
$\pi:\mathcal{X}\longrightarrow\mathcal{X}^{**}$ is the natural canonical
projection from $\mathcal{X}$ onto $\mathcal{X}^{**}.$ Also $T^{\dagger}$
denote the pseudo inverse of the operator $T.$ Note that $TT^{\dagger}f=f$ for
all $f\in R(K).$ Throughout $R(K)$ is closed.

######  Definition 1.1.

[15]Let $\mathcal{X}$ be a Banach space and $\mathcal{X}_{d}$ be a BK-space. A
sequence
$(x_{n},f_{n})(\\{x_{n}\\}\subset\mathcal{X},\\{f_{n}\\}\subset\mathcal{X}^{*})$
is called an _atomic decomposition_ for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ if the following statements hold:

  1. (a)

$\\{f_{n}(x)\\}\in\mathcal{X}_{d}$, for all $x\in~{}\mathcal{X}$.

  2. (b)

There exist constants $A$ and $B$ with $0<A\leq B<\infty$ such that

$\displaystyle
A\|x\|_{\mathcal{X}}~{}\leq~{}\|\\{f_{n}(x)\\}\|_{\mathcal{X}_{d}}~{}\leq~{}B\|x\|_{\mathcal{X}},~{}\text{for
all}\ x\in\mathcal{X}$ (1.2)

  3. (c)

$x=\sum\limits_{n=1}^{\infty}f_{n}(x)x_{n}$, for all $x\in\mathcal{X}$.

Next, we state some lemmas which we will use in the subsequent results.

######  Lemma 1.2.

[28, 31]Let $\mathcal{X}$, $\mathcal{Y}$ be Banach spaces and
$T:\mathcal{X}\longrightarrow\mathcal{Y}$ be a bounded linear operator. Then,
the following conditions are equivalent:

  1. (a)

There exist two continuous projection operators
$P:\mathcal{X}\rightarrow\mathcal{X}$ and
$Q:\mathcal{Y}\rightarrow\mathcal{Y}$ such that

$\displaystyle P(\mathcal{X})=kerT~{}and~{}Q(\mathcal{Y})=T(\mathcal{X}).$
(1.3)

  2. (b)

$T$ has a pseudo inverse operator $T^{\dagger}$.

If two continuous projection operators $P:\mathcal{X}\rightarrow\mathcal{X}$
and $Q:\mathcal{Y}\rightarrow\mathcal{Y}$ satisfies (2.3), then there exists a
pseudo inverse operator $T^{\dagger}$ of $T$ such that
$T^{\dagger}T=I_{\mathcal{X}}-P$ and $TT^{\dagger}=Q$, where $I_{\mathcal{X}}$
is the identity operator on $\mathcal{X}$.

######  Lemma 1.3.

[3, 24]Let $\mathcal{X}$ be a Banach space. If $T\in L(\mathcal{X})$ has a
generalized inverse S$\in L(\mathcal{X})$, then $TS$, $ST$ are projections and
$TS(\mathcal{X})=T(\mathcal{X})$ and $ST(\mathcal{X})=S(\mathcal{X})$.

######  Lemma 1.4.

[20, 26] Let $\mathcal{X}$ be a Banach space and
$\\{f_{n}\\}\subset\mathcal{X}^{*}$ be a sequence such that
$\\{x\in\mathcal{X}:f_{n}(x)=0,for\ all\ n\in\mathbb{N}\\}=\\{0\\}$. Then
$\mathcal{X}$ is linearly isometric to the Banach space
$\mathcal{X}_{d}=\\{\\{f_{n}(x)\\}:x\in\mathcal{X}\\}$, where the norm is
given by $\|\\{f_{n}(x)\\}\|_{\mathcal{X}_{d}}$=$\|x\|_{\mathcal{X}}$,
$x\in\mathcal{X}$.

##  2\. Main Results

Poumai and Jahan [23] defined and studied $K$-atomic decomposition as a
generalization of $K$-frames in Banach spaces. Here we shall extend this study
further and introduced the concept of approximative K-atomic decomposition in
Banach spaces and obtained new and interesting results. We start this section
with the following definition of approximative K-Atomic decomposition:

######  Definition 2.1.

Let $\mathcal{X}$ be a Banach Space,
$\\{x_{n}\\}\subset\mathcal{X},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*},$
where $\\{m_{n}\\}$ is an increasing sequence of positive integer and $K\in
L(\mathcal{X})$. A pair
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is called an _approximative K-atomic decomposition_ for $\mathcal{X}$ with
respect to $\mathcal{X}_{d},$ if the following statements holds:

  1. (a)

$\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\in\mathcal{X}_{d}$,
for all $x\in\mathcal{X}.$

  2. (b)

There exist constants $A$ and $B$ with $0<A\leq B<\infty$ such that

$\displaystyle A\parallel
K(x)\parallel_{\mathcal{X}}~{}\leq~{}\parallel\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\|_{\mathcal{X}_{d}}\leq
B\parallel x\parallel_{\mathcal{X}},~{}\mbox{for all}\ x\in\mathcal{X}.$

  3. (c)

$\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$
converges for all $x\in\mathcal{X}$ and
$K(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$.

The constants $A$ and $B$ are called lower and upper bounds of the
approximative $K$-atomic decomposition
$(\\{x_{n}\\},\\{h_{n,i}\\}_{i=1,2,3,...,m_{n}}).$

######  Observations.

If $(\\{x_{n}\\},\\{f_{n}\\})$ is a $K$-atomic decomposition for $\mathcal{X}$
with respect to $\mathcal{X}_{d},$ then for
$h_{n,i}=f_{i},i=1,2,...,n,n\in\mathbb{N},$ $(\\{x_{n}\\},\\{h_{n,i}\\})$ is
an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
some associated Banach space $\mathcal{X}_{d}.$

######  Remark 2.2.

Let $(x_{n},h_{n,i})$ be an approximative $K$-atomic decomposition for
$\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds $A$ and $B$.

### (I)

If $K=I_{\mathcal{X}}$, then $(x_{n},h_{n,i})$ is an approximative atomic
decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds
$A$ and $B$.

### (II)

If $K$ is invertible, then $(K^{-1}(x_{n}),h_{n,i})$ is an approximative
atomic decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$. In
the following example, we show the existence of approximative $K$-atomic
decomposition for a Banach space $\mathcal{X}$ with respect to an associated
BK space $\mathcal{X}_{d}$ .

######  Example 2.3.

Let $\mathcal{X}$ be a Banach Space. Let$\\{x_{n}\\}\subseteq\mathcal{X}$,
$\\{h_{n,i}\\}\subseteq\mathcal{X}^{*}$ such that
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{n}$ converges
for all $x\in\mathcal{X}$ and $x_{n}\neq 0,$ for all $n\in\mathbb{N}$. Also,
let
$\mathcal{X}_{d}=\\{\\{h_{n,i}\\}|\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}~{}\mbox{converges}\\}.$
Then $\mathcal{X}_{d}$ is a BK-space with norm
$\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}=\sup\limits_{1\leq
n<\infty}\parallel\sum\limits_{i=1}^{n}h_{n,i}x_{i}\parallel$. Define an
operator as $T:\mathcal{X}_{d}\longrightarrow\mathcal{X}$ as
$T\\{h_{n,i}\\}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
and define $S:\mathcal{X}\longrightarrow\mathcal{X}_{d}$ as
$S(x)=\\{h_{n,i}(x)\\},~{}x\in\mathcal{X}$. Take $K=TS.$ Then
$K:\mathcal{X}\longrightarrow\mathcal{X}$ is such that
$K(x)=TS(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},$
for all $x\in\mathcal{X},~{}i=1,2,...,n,~{}n\in\mathbb{N}.$ Clearly,
$\\{h_{n,i}(x)\\}\in\mathcal{X}_{d}$ and

$\displaystyle\|K(x)\|_{\mathcal{X}}$ $\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\bigg{\|}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}\bigg{\|}\leq\sup\limits_{1\leq
n<\infty}\bigg{\|}\sum\limits_{k=1}^{n}h_{k}(x)x_{k}\bigg{\|}$
$\displaystyle=$ $\displaystyle\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}\leq
C\parallel x\parallel_{\mathcal{X}},~{}\mbox{for all}~{}x\in\mathcal{X},$

where $C=\sup\limits_{1\leq n<\infty}\parallel S_{n}\parallel$ and
$S_{n}(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}.$
Hence, $(x_{n},h_{n,i})$ is an approximative $K$-atomic decomposition for
$\mathcal{X}$ with respect to $\mathcal{X}_{d}.$

In the following result, we give the characterization regarding the existence
of approximative $K$-atomic decompositions in Banach spaces.

######  Theorem 2.4.

Let $K\in L(\mathcal{X})$ with $K\neq 0.$ Then a Banach space $\mathcal{X}$
has an approximative $K$-atomic decomposition if and only if there exists a
sequence $\\{v_{i}\\}\subset B(\mathcal{X})$ of finite rank endomorphism such
that $K(x)=\sum\limits_{i=1}^{n}v_{i}(x),~{}~{}x\in\mathcal{X}.$

###### Proof.

Let $\\{x_{n}\\}\subset\mathcal{X}$ and
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*},$
where $\\{m_{n}\\}$ is an increasing sequence of positive integer such that
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an _approximative K-atomic decomposition_ for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$ Define

$\displaystyle S_{n}(x)=\sum_{i=1}^{m_{n}}h_{n,i}(x)x_{i},~{}~{}\mbox{for
all}~{}x\in\mathcal{X},n\in\mathbb{N}.$

Then for each $n\in\mathbb{N}$ and $x\in\mathcal{X}$, $S_{n}(x)$ is a well
defined continuous linear mapping on $\mathcal{X}$ such that
$\lim\limits_{n\to\infty}S_{n}(x)=x,~{}~{}x\in\mathcal{X}.$ Also by uniform
boundedness principle we have $\sup\limits_{1\leq
n\leq\infty}\|S_{n}(x)\|<\infty.$ Assume that $v_{1}=S_{1}$,
$v_{2n}=v_{2n+1}=\frac{1}{2}(S_{n+1}-S_{n})$,$n\in\mathbb{N}.$ Now, we compute

$\displaystyle\lim\limits_{n\to\infty}\sum_{i=1}^{n}v_{i}(x)$
$\displaystyle=\lim\limits_{n\to\infty}(S_{1}(x)+\frac{1}{2}(S_{2}(x)-S_{1}(x))+\frac{1}{2}(S_{2}(x)-S_{1}(x))$
$\displaystyle+\frac{1}{2}(S_{3}(x)-S_{2}(x))+\frac{1}{2}(S_{3}(x)-S_{2}(x))+...)$
$\displaystyle=\lim\limits_{n\to\infty}S_{n}(x)$
$\displaystyle=K(x),~{}~{}\mbox{for all}~{}~{}x\in\mathcal{X},K\in L(X).$

Therefore, $\lim\limits_{n\to\infty}\sum_{i=1}^{n}v_{i}(x)=K(\mathcal{X}).$
Conversely assume that there exists a sequence of finite rank endomorphism
$\\{S_{n}\\}\subset L(\mathcal{X})$ such that
$\lim\limits_{n\to\infty}S_{n}(x)=K(x),~{}~{}x\in\mathcal{X}.$ Then, each
$S_{n}(x)$ is of finite rank, there exist a sequence
$\\{y_{n,i}\\}_{i=m_{n-1}+1}^{m_{n}}\subset\mathcal{X}$ and a total sequence
of row finite matrix of functionals
$\\{g_{n,i}\\}_{i=m_{n-1}+1}^{m_{n}}\subset\mathcal{X^{*}}$ such that

$\displaystyle S_{n}(x)=\sum_{i=m_{n-1}+1}^{m_{n}}g_{n,i}(x)y_{n,i},\mbox{for
all}~{}~{}x\in\mathcal{X},~{}~{}n\in\mathbb{N}.$

Define sequences $\\{x_{n}\\}\subset\mathcal{X}$ and
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*},$
where $\\{m_{n}\\}$ is an increasing sequence of positive integers, by

$\displaystyle x_{i}=y_{n,i},~{}i=m_{n-1}+1,...m_{n};n=1,2,3...$

and

$\displaystyle
h_{n,i}=\begin{cases}0,~{}~{}~{}\mbox{for}~{}~{}~{}i=1,2,...,m_{n-1}\\\
g_{n,i},~{}~{}~{}\mbox{for}~{}~{}~{}i=m_{n-1}+1,...,m_{n}.\end{cases}$

Then $x_{n}\neq 0,$ so for each $x\in\mathcal{X}$ and $n\in\mathbb{N},$ we get

$\displaystyle\lim\limits_{n\to\infty}\sum_{i=1}^{m_{n}}h_{n,i}(x)x_{i}=\lim\limits_{n\to\infty}S_{n}(x)=K(x).$
(2.4)

Let $x\in\mathcal{X}$ be such that $h_{n,i}(x)=0,\mbox{for
all}~{}i=1,2,...m_{n},~{}~{}n\in\mathbb{N}.$ Then by equation (2.4) $K(x)=0.$
Thus by Lemma 1.4 there exist an associated Banach space
$\mathcal{X}_{d}=\\{\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},x\in\mathcal{X}\\}$
with norm given by
$\|\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\|_{\mathcal{X}_{d}}=\|x\|_{\mathcal{X}},~{}~{}\mbox{for
all}~{}x\in\mathcal{X}.$ Hence $(\\{h_{n,i}\\},\\{x_{n}\\})$ is an
approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$ ∎

Next, we give an example of an approximative $K$-atomic decomposition for
$\mathcal{X}$ which is not an approximative atomic decomposition for
$\mathcal{X}.$

######  Example 2.5.

Let $\mathcal{X}=c_{0}\ and\ \mathcal{X}_{d}=l_{\infty}$. Let
$\\{x_{n}\\}\subset\mathcal{X}$ be the sequence of standard unit vectors in
$\mathcal{X}$ and $\\{h_{n,i}\\}\subseteq\mathcal{X}^{*}$ be such that for
$x=\\{\alpha_{n}\\}\in\mathcal{X},h_{n,1}(x)=0,h_{n,2}(x)=\alpha_{2},...,h_{n,i}(x)=\alpha_{n},....$
It is clear that
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$ converges
for $x\in\mathcal{X}$. Define $K:\mathcal{X}\longrightarrow\mathcal{X}$ by
$K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$ ,
$x\in\mathcal{X}.$ Then $\\{h_{n,i}(x)\\}\in\mathcal{X}_{d}$ is such that
$(\\{x_{n}\\},\\{h_{n,i}(x)\\})$ is an approximative $K$-atomic decomposition
for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$. But
$(\\{x_{n}\\},\\{h_{n,i}\\})$ is not an approximative atomic decomposition for
$\mathcal{X}.$

Next, we give various methods for the construction of approximative $K$-atomic
decompositions for $\mathcal{X}.$

######  Theorem 2.6.

Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B$. Let $K\in L(\mathcal{X})$ with
$K\neq 0$. Then
$(\\{Kx_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{i=1,2,3,...,m_{n}})$ is
an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $\frac{A}{\|K\|}$ and $B.$

###### Proof.

Since
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B.$ So for each $x\in\mathcal{X},$ we
have $x=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}.$
This implies
$K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)K(x_{i}).$
Also, we have $\parallel K(x)\parallel_{\mathcal{X}}~{}\leq~{}\parallel
K\parallel\parallel x\parallel_{\mathcal{X}}$, for all $x\in\mathcal{X}.$ This
gives

$\displaystyle\dfrac{A}{\parallel K\parallel}\parallel
K(x)\parallel_{\mathcal{X}}~{}\leq~{}\parallel\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}\leq
B\parallel x\parallel_{\mathcal{X}},~{}\mbox{for all}\ x\in\mathcal{X}.$

∎

######  Theorem 2.7.

Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B$. Let $K\in L(\mathcal{X})$ with
$K\neq 0$. Then
$(\\{x_{n}\\},\\{K^{*}h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B\|K\|.$

###### Proof.

Can be easily proved. ∎

######  Theorem 2.8.

Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B$ and let $T\in L(\mathcal{X})$ with
$T\neq 0$. Then $(\\{Tx_{n}\\},\\{h_{n,i}\\})$ is an approximative $TK$-atomic
decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds
$\frac{A}{\|T\|}$ and $B.$

###### Proof.

Construction of proof is similar to theorem 2.6. ∎

######  Theorem 2.9.

Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B$ and let $T\in L(\mathcal{X})$ with
$\|T\|\neq 0$ Then $(\\{x_{n}\\},\\{T^{*}h_{n,i}\\})$ is an approximative
$KT$-atomic decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$
with bounds $A$ and $B\|T\|.$

###### Proof.

Obvious ∎

######  Theorem 2.10.

If
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ and $K$ has pseudo inverse $K^{\dagger}$, then there exists
$\\{g_{n,i}\\}\subseteq\mathcal{X}^{*}$ such that
$(\\{x_{n}\\},\\{g_{n,i}\\})$ is an approximative $K$-atomic decomposition for
$\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds $A$ and
$B\|K\|^{2}$

###### Proof.

Since
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d},$ then for each $x\in\mathcal{X}$ we have

$\displaystyle
A\|K(x)\|_{\mathcal{X}}\leq\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}\leq
B\|x\|_{\mathcal{X}},~{}x\in\mathcal{X}.$

Also, for each $x\in\mathcal{X}$, we have

$\displaystyle
K(x)=K(K^{\dagger}K(x))=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(K^{\dagger}K(x))x_{i}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}((K^{\dagger}K)^{*}(h_{n,i})(x))x_{i}.$

For each $n\in\mathbb{N}$, define $g_{n,i}={(K^{\dagger}K)}^{*}(h_{n,i})$.
Then

$\|K(x)\|_{\mathcal{X}}=\|K(K^{\dagger}K(x))\|_{\mathcal{X}}\leq\dfrac{1}{A}\|\\{h_{n,i}(K^{\dagger}K(x))\\}\|_{\mathcal{X}_{d}}=\dfrac{1}{A}\|\\{g_{n,i}(x)\\}\|_{\mathcal{X}_{d}},~{}x\in\mathcal{X}$

and

$\|\\{g_{n,i}(x)\\}\|_{\mathcal{X}_{d}}=\|\\{h_{n,i}(K^{\dagger}K(x))\\}\|_{\mathcal{X}_{d}}\leq
B\|K^{\dagger}\|\|K\|\|x\|_{\mathcal{X}},~{}x\in\mathcal{X}.$

Hence, we conclude that $(\\{x_{n}\\},\\{g_{n,i}\\})$ is an approximative
$K$-atomic decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$.
∎

##  3\. Approximative $\mathcal{X}_{d}$-frame

Casazza et al. [5] defined and studied $\mathcal{X}_{d}$-Bessel sequences and
$\mathcal{X}_{d}$-frames in Banach spaces. Later on Stoeva [27] studied
perturbation of $\mathcal{X}_{d}$-Bessel sequences, $\mathcal{X}_{d}$-frames,
atomic decomposition and $\mathcal{X}_{d}$-Riesz bases in separable Banach
spaces. We have generalized this concept and defined approximative
$\mathcal{X}_{d}$-Bessel sequences and approximative $\mathcal{X}_{d}$-frames
in Banach spaces. We begin this section with the following definitions:

######  Definition 3.1.

A sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*},$
where $\\{m_{n}\\}$ is an increasing sequence of positive integers, is called
an approximative _$\mathcal{X}_{d}$ -frame_ for $\mathcal{X}$ if

  1. (a)

$\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\in\mathcal{X}_{d}$,
for all $x\in\mathcal{X}.$

  2. (b)

There exist constants $A$ and $B$ with $0<A\leq B<\infty$ such that

$\displaystyle
A\|x\|_{\mathcal{X}}~{}\leq\|\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\|_{\mathcal{X}_{d}}~{}\leq
B\|x\|_{\mathcal{X}},~{}\text{for all}\ x\in\mathcal{X}.$ (3.5)

The constants $A$ and $B$ are called approximative _$\mathcal{X}_{d}$ -frame_
bounds. If atleast (a) and the upper bound condition in (3.5) are satisfied,
then $\\{h_{n,i}\\}$ is called an approximative _$\mathcal{X}_{d}$ -Bessel
sequence_ for $\mathcal{X}.$
One may note that if $\\{f_{n}\\}$ is an $\mathcal{X}_{d}$-frame for
$\mathcal{X},$ then for
$\\{h_{n,i}\\}=f_{i},~{}i=1,2,3,...,n;~{}n\in\mathbb{N}$, $\\{h_{n,i}\\}$ is
an approximative $\mathcal{X}_{d}$-frame for $\mathcal{X}.$ Also, note that if
$\\{f_{n}\\}$ is an $\mathcal{X}_{d}$-Bessel sequence for $\mathcal{X},$ then
for $\\{h_{n,i}\\}=f_{i},~{}i=1,2,3,...,n;~{}n\in\mathbb{N}$, $\\{h_{n,i}\\}$
is an approximative $\mathcal{X}_{d}$-Bessel sequence for $\mathcal{X}.$
In the next two results, we give necessary conditions under which an
approximative $\mathcal{X}_{d}$-frame gives rise to a bounded operator $K$
with respect to which there is an approximative $K$-atomic decomposition for
$\mathcal{X}$.

######  Theorem 3.2.

Let
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*}$
be an approximative $\mathcal{X}_{d}$-frame for $\mathcal{X}$ with bounds $A$
and $B$. Let $\\{x_{n}\\}\subseteq\mathcal{X}$ with $\sup\limits_{1\leq
n<\infty}\|x_{n}\|<\infty$ and let
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}|h_{n,i}(x)|<\infty,$ for
all $x\in\mathcal{X}$. Then there exists an operator $K\in L(\mathcal{X})$
such that $(\\{x_{n}\\},\\{h_{n,i}\\})$ is an approximative $K$-atomic
decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$.

###### Proof.

Since
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*}$
is an approximative $\mathcal{X}_{d}$-frame for $\mathcal{X}$ with
$\sup\limits_{1\leq n<\infty}\|x_{n}\|<\infty$ and
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}|h_{n,i}(x)|<\infty.$ Then,
by Theorem 2.4, we have
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$ exist for
all $x\in\mathcal{X},~{}n\in\mathbb{N}.$
Define $K:\mathcal{X}\longrightarrow\mathcal{X}$ by
$K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$,
$x\in\mathcal{X}$. Then $K$ is a bounded linear operator such that

$\|K(x)\|_{\mathcal{X}}\leq\sup\limits_{1\leq
n<\infty}\|\sum_{i=1}^{m_{n}}h_{n,i}(x)x_{i}\|_{\mathcal{X}}\leq
C\|x\|_{\mathcal{X}},$

where $C=\sup\limits_{1\leq
n<\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$. Thus

$\dfrac{A}{C}\|K(x)\|_{\mathcal{X}}\leq\parallel\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}\leq
B\parallel x\parallel_{\mathcal{X}},~{}\mbox{for all}~{}x\in\mathcal{X}.$

Hence, $(\\{x_{n}\\},\\{h_{n,i}\\})$ is an approximative $K$-atomic
decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds
$\dfrac{A}{C}$ and $B$. ∎

######  Theorem 3.3.

Let
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*}$
be an approximative $\mathcal{X}_{d}$-frame with bounds $A$, $B$ and let
$\\{x_{n}\\}\subseteq\mathcal{X}$. Let
$T:\mathcal{X}_{d}\longrightarrow\mathcal{X}$ given by
$T(\\{h_{n,i}\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
be a well defined operator. Then, there exists a linear operator $K\in
L(\mathcal{X})$ such that $(\\{x_{n}\\},\\{h_{n,i}\\})$ is an approximative
$K$-atomic decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}.$

###### Proof.

Define $U:\mathcal{X}\longrightarrow\mathcal{X}_{d}$ by
$U(x)=\\{h_{n,i}(x)\\}$, $x\in\mathcal{X}$. Then $U$ is well defined and
$\|U\|\leq B$. Take $K=TU$. Then
$K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},~{}x\in\mathcal{X}$.
Therefore, by uniform boundedness principle, we have

$\|K(x)\|_{\mathcal{X}}\leq\sup\limits_{1\leq
n<\infty}\|\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}\|_{\mathcal{X}}\leq
C\|x\|_{\mathcal{X}},~{}x\in\mathcal{X},$

where $C=\sup\limits_{1\leq
n<\infty}\|\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}\|_{\mathcal{X}}$. Thus, we
have

$\dfrac{A}{C}\|K(x)\|~{}\leq~{}\parallel\\{h_{n,i}(x)\\}\|\leq B\parallel
x\parallel,~{}\mbox{for all}\ x\in\mathcal{X}.$

Hence $(\\{x_{n}\\},\\{h_{n,i}\\})$ is an approximative $K$-atomic
decomposition for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$ with bounds
$\dfrac{A}{C}$ and $B$. ∎

Next, we give the existence of an approximative $K$-atomic decomposition from
an approximative $\mathcal{X}_{d}$ Bessel sequence.

######  Theorem 3.4.

Let $\mathcal{X}$ be a reflexive Banach space and $\mathcal{X}_{d}$ be a BK-
space which has a sequence of canonical unit vectors $\\{e_{n}\\}$ as a basis.
Let
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*}$
be an approximative $\mathcal{X}_{d}$-Bessel sequence with bound $B$ and let
$\\{x_{n}\\}\subseteq\mathcal{X}$. If $\\{h(x_{n})\\}\in(\mathcal{X}_{d})^{*}$
for all $h\in\mathcal{X}^{*}$, then there exists a bounded linear operator
$K\in L(\mathcal{X})$ such that
$(\\{x_{n}\\},~{}\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$

###### Proof.

Clearly $U:\mathcal{X}\longrightarrow\mathcal{X}_{d}$ given by
$U(x)=\\{h_{n,i}(x)\\},~{}x\in\mathcal{X}$ is well defined. Define a map
$R:\mathcal{X}^{*}\longrightarrow(\mathcal{X}_{d})^{*}$ by
$R(h)=\\{h(x_{n})\\},~{}x\in\mathcal{X}$. Then, its adjoint
$R^{*}:(\mathcal{X}_{d})^{**}\longrightarrow\mathcal{X}^{**}$ is given by
$R^{*}(e_{j})(h)=e_{j}(R(h))=h(x_{j})$. Let $T=(R^{*})|_{\mathcal{X}_{d}}$ and
$\\{h_{n,i}\\}\in\mathcal{X}_{d}$. Then

$\displaystyle
T(\\{h_{n,i}\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}T(e_{i})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}.$

But $\\{h_{n,i}(x)\\}\in\mathcal{X}_{d}$. So
$T(\\{h_{n,i}(x)\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$.
Take $K=TU$. Then $K\in L(\mathcal{X})\ \text{and}\
K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}.$
Moreover, $T$ is a bounded linear operator such that
$\|K(x)\|\leq\|T\|\|\\{h_{n,i}(x)\\}\|$. Hence

$\dfrac{1}{\|T\|}\|K(x)\|\leq\|\\{h_{n,i}(x)\\}\|\leq
B\|x\|,~{}x\in\mathcal{X}$

∎

Next, we construct an approximative $K^{*}$-atomic decomposition for
$\mathcal{X}^{*}$ from a given approximative $K$-atomic decomposition for
$\mathcal{X}.$

######  Theorem 3.5.

Let $\mathcal{X}_{d}$ be a BK-space with dual $(\mathcal{X}_{d})^{*}$ and let
$\mathcal{X}_{d}$ and $(\mathcal{X}_{d})^{*}$ have sequences of canonical unit
vectors $\\{e_{n}\\}\ \text{and}\ \ \\{v_{n}\\}$ respectively as bases. Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$. Let $S:\mathcal{X}_{d}\longrightarrow\mathcal{X}$ given by
$S(\\{h_{n,i}\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
be a well defined mapping. Then,
$(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},\pi(x_{n}))$
is an approximative $K^{*}$-atomic decomposition for $\mathcal{X}^{*}$ with
respect to $(\mathcal{X}_{d})^{*}$.

###### Proof.

Since
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d},$ so for each $x\in\mathcal{X}$,
$K(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}.$ Thus
$h(K(x))=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)h(x_{i}).$
Therefore, by Theorem 2.4 we have
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}$ exist for
all $h\in\mathcal{X}^{*}.$ Also, for $x\in\mathcal{X}$, we compute

$(K^{*}(h))(x)=h(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}(x).$

This gives
$K^{*}(h)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}$,
for $h\in\mathcal{X}^{*}$. Note that
$S^{*}(h)(e_{j})=h(S(e_{j}))=h(x_{j}),h\in\mathcal{X}^{*}$. So,
$S^{*}(h)=\\{h(x_{n})\\}$ and
$\\{h(x_{n})\\}=\\{h(S(e_{n}))\\}\in(\mathcal{X}_{d})^{*},h\in\mathcal{X}^{*}$.
Also

$\displaystyle\|\\{h(x_{n})\\}\|_{(\mathcal{X}_{d})^{*}}=\|S^{*}(h)\|\leq\|S\|\|h\|_{\mathcal{X}^{*}},~{}h\in\mathcal{X}^{*}.$

Define $R:\mathcal{X}\longrightarrow\mathcal{X}_{d}$ by
$R(x)=\\{h_{n,i}(x)\\},x\in\mathcal{X}$. Then,
$R^{*}(v_{j})(x)=v_{j}(R(x))=h_{j,i}(x)$, $x\in\mathcal{X}$. So,
$R^{*}(v_{j})=h_{j,i}$, for all $j\in\mathbb{N}$ and for
$\\{g_{n,i}\\}\in(\mathcal{X}_{d})^{*}$ we have

$\displaystyle
R^{*}(\\{g_{n,i}\\})=R^{*}(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}g_{n,i}(x)v_{i})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}g_{n,i}(x)h_{n,i}.$

Therefore, we have

$\displaystyle
R^{*}S^{*}(h)=R^{*}(\\{h(x_{i})\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i},~{}h\in\mathcal{X}^{*}.$

Note that, $K^{*}=R^{*}S^{*}$ and so

$\displaystyle\|K^{*}(h)\|_{\mathcal{X}^{*}}=\|R^{*}S^{*}(h)\|_{\mathcal{X}^{*}}\leq\|R^{*}\|\|\\{h(x_{n})\\}\|_{(\mathcal{X}_{d})^{*}},~{}h\in\mathcal{X}^{*}.$

This gives

$\dfrac{1}{\|R^{*}\|}\|K^{*}(h)\|_{\mathcal{X}^{*}}\leq\|\\{h(x_{n})\\}\|_{(\mathcal{X}_{d})^{*}}\leq\|S\|\|h\|_{\mathcal{X}^{*}},~{}h\in\mathcal{X}^{*}.$
(3.6)

Hence,
$(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},\pi(x_{n}))$
is an approximative $K^{*}$-atomic decomposition for $\mathcal{X}^{*}$ with
respect to $(\mathcal{X}_{d})^{*}$. ∎

Next, we give the following result characterizing the class of approximative
$K$-atomic decompositions.

######  Theorem 3.6.

Let
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
be an approximative $K$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ with bounds $A$ and $B$. Let
$T:\mathcal{X}_{d}\longrightarrow\mathcal{X}$ given by
$T(\\{h_{n,i}\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
is well defined for $\\{h_{n,i}\\}\in\mathcal{X}_{d}$ and let
$U:\mathcal{X}\longrightarrow\mathcal{X}_{d}$ be the mapping given by
$U(x)=\\{h_{n,i}(x)\\}$. If $K$ is invertible, then the following statements
are equivalent.

  1. (a)

$T$ is the pseudo inverse of $U$.

  2. (b)

$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$

  3. (c)

$T$ is a linear extension of
$U^{-1}:U(\mathcal{X})\longrightarrow\mathcal{X}$.

  4. (d)

$U(\mathcal{X})$ is a complemented subspace of $\mathcal{X}_{d}.$

  5. (e)

$KerT$ is a complemented subspace of $\mathcal{X}_{d}$ and $T$ is surjective.

###### Proof.

$(a)\Rightarrow(b)$ By hypothesis, $\\{x\in\mathcal{X}:h_{n,i}(x)=0,~{}for\
all\ n\in\mathbb{N}\\}=\\{0\\}$. So, $KerU=\\{0\\}$. Since $T$ is the pseudo
inverse of $U$, by Lemma 1.2 there exists a continuous projection operator
$\theta:\mathcal{X}\longrightarrow\mathcal{X}$ such that
$TU=I_{\mathcal{X}}-\theta$ and $kerU=\theta(\mathcal{X})$. Thus, for each
$x\in\mathcal{X},$ we have

$\displaystyle TU(x)=(I_{\mathcal{X}}-\theta)(x)=x,~{}x\in\mathcal{X}.$

Hence, for every $x\in\mathcal{X}$,
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}=x$.
$(b)\Rightarrow(a)\ $For $x\in\mathcal{X}$, we have

$\displaystyle
UTU(x)=UT(\\{h_{n,i}(x)\\})=U(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i})=U(x).$

Hence, $UTU=U$.
$(c)\Rightarrow(b)\ \ $If $T$ is a linear extension of
$U^{-1}:U(\mathcal{X})\longrightarrow\mathcal{X}$, then
$TU:\mathcal{X}\longrightarrow\mathcal{X}$ is the identity map on
$\mathcal{X}$. So, $TU(x)=x$ and
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}=x$.
$(c)\Rightarrow(a)\ $Obvious, since $UTU=UI_{\mathcal{X}}=U.$
(d)$\Rightarrow$(b) Suppose $\mathcal{X}_{d}=U(\mathcal{X})\oplus G$, where
$G$ is a closed subspace of $\mathcal{X}_{d}$. Let _P_ be a projection of
$\mathcal{X}_{d}$ onto $U(\mathcal{X})$ along $G$.
Then,
$\emph{P}(\\{h_{n,i}\\})=\\{g_{n,i}(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i})\\}$,
for all $\\{h_{n,i}\\}\in\mathcal{X}_{d}$. Therefore

$\displaystyle U^{-1}\circ\emph{P}(\\{h_{n,i}\\})$ $\displaystyle=$
$\displaystyle
U^{-1}\\{g_{n,i}(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i})\\}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
$\displaystyle=$ $\displaystyle T(\\{h_{n,i}\\}),~{}for\ all\
\\{h_{n,i}\\}\in\mathcal{X}_{d}.$

This gives, T=$U^{-1}\circ\emph{P}$ and

$\displaystyle
T(\\{h_{n,i}(x)\\})=U^{-1}\circ\emph{P}(\\{h_{n,i}(x)\\})=U^{-1}(\\{h_{n,i}(x)\\}.$

Hence, $x=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$,
for all $x\in\mathcal{X}$.
(b)$\Rightarrow$(d) Obvious.
(e)$\Rightarrow$(b) Let $\mathcal{X}_{d}=kerT\oplus M$, where $M$ is a closed
subspace of $\mathcal{X}_{d}.$ Take $\Upsilon=kerT\oplus U(\mathcal{X}).$ Let
$Q:\mathcal{X}_{d}\longrightarrow M$ be a projection from $\mathcal{X}_{d}$
onto $M$ along $kerT$. Define $L:\mathcal{X}_{d}\longrightarrow\Upsilon$ by
$L(\alpha)=(\alpha-Q(\alpha),UT(\alpha))$, for
$\alpha=\\{h_{n,i}\\}\in\mathcal{X}_{d}.$ Let $L(\alpha)=0$. This gives
$Q(\alpha)=\alpha.$ So $\alpha\in M.$ Let $UT(\alpha)=0.$ Then

$\displaystyle
U(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i})=\\{g_{n,i}(\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i})\\}=0,~{}for\
n\in\mathbb{N}.$

This gives $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}=0$
and so, $\alpha\in kerT$. Thus, $\alpha\in kerT\cap M=\\{0\\}$. Hence, $L$ is
one-one.
Let $(\alpha_{0},U(x))\in kerT\oplus U(\mathcal{X}),$ for $\alpha_{0}\in kerU$
and $U(x)\in U(\mathcal{X}).$
Since, $T$ is onto, for each $x\in\mathcal{X}$, there exists
$\beta\in\mathcal{X}_{d}$ such that $T(\beta)$ = $x$ and this gives
$UT(\beta)=U(x).$ Take $\alpha=\alpha_{0}+Q(\beta)$. Then
$Q(\alpha)=Q(\alpha_{0})+Q^{2}(\beta)=Q(\beta)$ and
$\alpha_{0}=\alpha-Q(\alpha)$. Also, we have

$\displaystyle UT(\alpha)=UT(\alpha-\alpha_{0})=UT(Q(\beta))=UT(\beta)=U(x).$
(3.7)

Thus $L(\alpha)=(\alpha_{0},UT(x))$ and $L$ is an isomorphism from
$\mathcal{X}_{d}$ onto $\Upsilon.$ So, there is a projection
$\emph{P}=UT:\mathcal{X}_{d}\longrightarrow U(\mathcal{X})$ onto
$U(\mathcal{X})$ along $kerT$. This gives

$\displaystyle U^{-1}\circ\emph{P}=T\ \ and\ \
U^{-1}\circ\emph{P}(\\{h_{n,i}(x)\\})=T(\\{h_{n,i}(x)\\}).$

Finally, we compute

$\displaystyle
U^{-1}(\\{h_{n,i}(x)\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}\
\ and\ \ x=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}.$

Therefore,
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$.
(b)$\Rightarrow$(e) Obvious. ∎

In the following result, we prove a duality type approximative $K$-atomic
decomposition for $\mathcal{X}$.

######  Theorem 3.7.

Let $\mathcal{X}_{d}$ be a reflexive BK-space with its dual
$(\mathcal{X}_{d})^{*}$ and let sequences of canonical unit vectors
$\\{e_{n}\\}\ and\ \\{v_{n}\\}$ be bases for $\mathcal{X}_{d}$ and
$(\mathcal{X}_{d})^{*}$, respectively. Let
$(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},\pi(x_{n}))$
be an approximative $K$-atomic decomposition for $\mathcal{X}^{*}$ with
respect to $(\mathcal{X}_{d})^{*}.$ If
$S:({\mathcal{X}_{d}})^{*}\longrightarrow\mathcal{X}^{*}$ given by
$S(\\{d_{i}\\})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}d_{i}h_{n,i}$
is well defined for $\\{d_{i}\\}\in\mathcal{X}_{d}^{*}$, then there exists a
linear operator $L\in L(\mathcal{X})$ such that
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $L$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$

###### Proof.

Since
$(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},\pi(x_{n}))$
is an approximative $K$-atomic decomposition for $\mathcal{X}^{*}$ with
respect to $(\mathcal{X}_{d})^{*}.$ For $h\in\mathcal{X}^{*}$, we have
$K(h)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}$. Also,
by Theorem 2.4 we have
$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$ exist, for
all $x\in\mathcal{X}.$ Define$L:\mathcal{X}\longrightarrow\mathcal{X}$ by
$L(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},~{}x\in\mathcal{X}.$
Note that $S(v_{n})=h_{n,i},~{}n\in\mathbb{N}$ and for $x\in\mathcal{X}$, the
linear bounded operator
$S^{*}:\mathcal{X}^{**}\longrightarrow(\mathcal{X}_{d})^{**}$ satisfies

$\displaystyle S^{*}(\pi(x))(v_{n})=\pi(x)S(v_{n})=h_{n,i}(x).$

So, $\\{h_{n,i}(x)\\}$ is identified with
$S^{*}(\pi(x))\in(\mathcal{X}_{d})^{**}=\mathcal{X}_{d}$. Further, we have

$\displaystyle\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}=\|S^{*}(\pi(x))\|_{\mathcal{X}_{d}}\leq\parallel
S\parallel\parallel x\parallel_{\mathcal{X}},~{}x\in\mathcal{X}.$ (3.8)

Letting $U={S^{*}}\mid_{\mathcal{X}}$, we have $U(x)$ =$\\{h_{n,i}(x)\\}$ and
$\|U\|\leq\|S\|$.
Define $R:\mathcal{X}^{*}\longrightarrow(\mathcal{X}_{d})^{*}$ by
$R(f)=\\{h(x_{n})\\},h\in\mathcal{X}^{*}$. Then

$\displaystyle R^{*}(e_{j})(h)=e_{j}(R(h))=h(x_{j}),~{}\ h\in\mathcal{X}^{*}.$

So, $R^{*}(e_{j})=x_{j},\mbox{for all}\ j\in\mathbb{N}$. Take
$T=(R^{*})|_{\mathcal{X}_{d}}$. Then, for $\\{h_{n,i}\\}\in\mathcal{X}_{d}$ we
compute

$\displaystyle
T(\\{h_{n,i}\\})=T(h_{n,i}e_{i})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}T(e_{i})=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}.$

Thus,
$TU(x)=\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$, for
all $x\in\mathcal{X}$ and this gives $TU=L$ on $\mathcal{X}.$ Therefore,
$\dfrac{1}{\|T\|}\|L(x)\|_{\mathcal{X}}\leq\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}.$
Then

$\displaystyle\dfrac{1}{\|T\|}\|L(x)\|_{\mathcal{X}}\leq\|\\{h_{n,i}(x)\\}\|_{\mathcal{X}_{d}}\leq\|S\|\|x\|_{\mathcal{X}}.$

Hence,
$(\\{x_{n}\\},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative $L$-atomic decomposition for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}.$ ∎

##  4\. Possible Application

One of the most important device in modern world is digital camera. In our
notation a digital picture is a two-dimensional sequence, $\\{h_{nm}\\}.$ So,
it can be seen either as an infinite length sequence with a finite number of
non-zeros samples; that is $\\{h_{nm}\\},~{}~{}n,m\in\mathbb{Z},$ or as a
sequence with domain $n\in\\{0,1,2,...,N-1\\},$ $m\in\\{0,1,2,...,M-1\\},$ can
be expressed as a matrix:

$h=\begin{bmatrix}h_{0,0}&h_{0,1},&.&.&.,&h_{M-1}\\\
h_{1,0}&h_{1,1},&.&.&.,&h_{M-1}\\\
h_{N-1,0}&h_{N-1,1},&.&.&.,&h_{N-1,M-1};\end{bmatrix}$

where each elements $h_{nm}$ is called a pixel and the image has $NM$ pixels.
In real life for $h_{n,m}$ to represent colour image, it must have more than
one component, usually, red, green and blue components are used(RGB colour
space).

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