Vol 5, No 3 (2014)

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Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 5(3), March – 2014 129

A NOTE ON

X

_d

-FRAMES IN BANACH SPACES

Ram Bharat Singh

1

_{, Mahesh C. Joshi}

1

**_{and A. K. Sah*}**

2

1

_{Department of Mathematics, Kumaun University, D. S. B. Campus, Nainital-263001, India.}

2

_{Department of Mathematics, Kirori Mal College, University of Delhi, Delhi-110007, India.}

(Received on: 30-01-14; Revised & Accepted on: 07-03-14)

ABSTRACT

F

rames with respect to some sequence space, namely

X

_d-frames were introduced and studied by Casazza et al [1]. We further study

X

_d-frames and proved if a Banach space has a

X

_d-frames then it also has the Parseval and an exact Parseval

X

_d-frames. A necessary and sufficient condition and also a sufficient condition for the stability of

d

X

-frame have been given. Also,

X

_d-Bessel sequences, for a sequence to be

X

_d -frame has been obtained.

2000 AMS Subject Classification: 42C15, 42A38.

Key Words:

X

_d -frame,

X

_d -Bessel sequence, Banach frame, Stability, an exact

X

_d -frame.

1. INTRODUCTION

Fourier transform has been a major tool in analysis for over a century. It has a lacking for signal analysis in which it hides in its phases information concerning the moment of emission and duration of a signal. What was needed was localized time frequency representation which has this information encoded in it. In 1946 Dennis Gabor [8], filled this gap and formulated a fundamental approach to signal decomposition in terms of elementary signals. On the basis of this development, in 1952 the notion of frame was determined by Duffin and Schaeffer [6] in Hilbert spaces in the

following way: Let X be a Separable Hilbert space, the system of non-zero elements

{ }

x

n n

⊂

X

be called a frame in X if there exist the constants

0 <

A

≤

B

< ∞

such that for each

x



X

,

it is valid

2 2 2

1

| ( ,

) |

X n X

n

A

x

x x

B

x

∞

=

≤

∑

≤

 

(1.1)

where

 

•

_X and

( , )

• •

is a norm and scalar product in

X

respectively. The constants A and B in

(1.1)

are called lower and upper frame bounds, the number

K

=

A B

/

is called condition coefficient of the frame

{ }

x

n n. In the case, when

K

=

1

,

{ }

x

_{n n}_ is a tight frame. Development in theory of frames in Hilbert space reduced to obtaining the analogues of the known results for the Banach case. By the theory of frames [3, 6, 7, 9] we have their Banach extensions in [1, 2, 4, 5, 10]. Frames have many properties of bases but lacks a very important one, namely, uniqueness. This property of frames make them very useful in the study of function spaces, signal and image processing, filter banks, wireless and communications etc.

The notion of

X

_d -frame generalizing the notion of

p

-frame studied in [4] was introduced in [1]. In the present paper, we further study

X

_d -frames and obtain a necessary and sufficient condition. Also, a sufficient condition for stability of

X

_d -frame has been given and proved if a Banach spaces has a

X

_d-frame then it is also has a Parseval and an exact Parseval

X

_d -frames.

Further,

X

_d -Bessel sequence has been studied and a sufficient condition in terms of

X

_d-Bessel sequence, for a sequence to be a

X

_d-frame has been given.

Corresponding author: A. K. Sah*

2

_{Department of Mathematics, Kirori Mal College, University of Delhi, Delhi-110007, India.}

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2. PRELIMINARIES

Throughout this paper

X

will denote an infinite dimensional Banach space over the scalar field



(



or



). Let

X

∗

and

X

∗∗denote the first and second conjugate space of

X

respectively,

L X X

( ,

)

Banach space of all continuous linear mapping from

X

into

X

, and

X

_d be an associated Banach space of scalar valued sequences index by



. Let

[

f

_n

]

the closed linear span of

{ }

f

_n and

[



f

_n

]

the closed linear span of

{ }

f

_n in

σ

(

X

*

,

X

)

-topology. A sequence

*

{ }

f

n

⊂

X

is said to be complete if

[

f

n

]

=

X

*and total if {

x



X

:

f

n

( )

x

=

0,

n

 

}

=

{0}

.

Definition: 2.1 ([1]) A sequence space

X

_d is called a BK-space, if it is a Banach space and the coordinate functional are continuous on

X

_d, i.e., the relations

x

_n

=

{ }

α

_j( )n ,

x

=

{ }

α

_j

∈

X

_d,

lim

n

n→∞

x

=

x

imply

( )

lim

_jn _j

(

1, 2,...).

n→∞

α

=

α

j

=

Definition: 2.2 ([1]) Let

X

_d be a BK-space and

{ }

f

_{i i}∞₌₁

⊂

X

∗. The sequence

{ }

f

_{i i}∞₌₁ is called a

X

_d -frame for

X

with lower bound

A

and upper bound

B

if

0 <

A

≤

B

< ∞

and for every

x

∈

X

one has (i)

{ ( )}

f x

i i∞=₁



X

d;

(ii)

{ ( )}

₁

d

i i

X _X X

A x

≤

f x

∞₌

≤

B x

. (2.1)

When (i) and the upper inequality in (ii) hold for every

x

∈

X

,

{ }

f

i i∞=₁ is called a

X

d -Bessel sequence for

X

with bound

B

. The positive constants

A

and

B

, respectively, are called lower and upper frame bounds of the

X

_d-frame. The inequality (2.1) is called the frame inequality. It is easy to observe that frame bounds need not be unique. Further,

the

X

_d-frame is called tight frame if it is possible to choose

A

and

B

, satisfying (2.1) with

A

=

B

and normalized tight (or Parseval) if

A

=

B

=

1

. If removal of one

f

_n renders the collection

{ }

f

_n

⊂

X

*no longer a

X

_d-frame for

X

, then { }

f

n is called can an exact

X

d -frame.

Let

{ }

f

_{i i}∞₌₁

⊂

X

∗. The operators

U

and

T

given by

Ux

=

{

f x

_i

( ) ,

}

x

∈

X

,

and

{ }

0

i i i

i

T d

d f

∞

=

∑

, are called the

analysis operator for

{ }

f

_{i i}∞₌₁ and the synthesis operator for

{ }

f

_{i i}∞₌₁, respectively.

3. MAIN RESULTS

The following result which is referred in this paper is listed in the form of a lemma.

Lemma: 3.1 ([11]) If

{ }

f

n

⊂

X

∗ is a

X

d-frame for

X

with respect to

X

d. Then

{ }

f

n is an exact iff



[ ]

n i i n

f

∉

f

_≠ , for all

n

.

Proof: It is straight forward on the line of totalness of

{ }

f

_n obtained from frame inequality of

X

_d -frame.

Theorem: 3.2 If

X

be a Banach space having a

X

_d -frame. Then

X

has a Parseval

X

_d- frame as well as exact Parseval

X

_d-frame.

Proof: Let

{ }

f

_n

⊂

X

∗ be a

X

_d -frame for

X

with respect to

X

_d. Then by frame inequality, { }

f

_n is total over

X

.

Therefore by Remark 7.1, in [12], there exists an associated Banach space {{

f

n

( )} :

x



X

}

with norm given by

{

( )}

d

n X X

f

x

=

x

,

x



X

.

(3)

{ }

f

_n is finitely linear independent. (In case

{ }

f

_n is not finitely linear independent, we can derive a subsequence

{ }

g

_i

⊂

{ }

f

_n which is finitely linear independent and total over

X

.) Then, for each

n

 

, there exists an element

x

_n



X

, such that

f x

_i

(

_n

)

=

0

,

i

=

1, 2,...,

n

−

1

, and

(

)

1

n n

f

x

=

, indeed if

f x

_i

( )

=

0 (

i

=

1, 2,...

n

−

1)

would imply

f x

_n

( )

=

0

, then we would have by [4],

1 2

[ ,

,...,

]

n n

g

∈

g g

g

, contradicting our assumption. Define

{ }

h

n

⊂

X

∗ by

h

1

=

f

1 and 1

1

( ) ,

2, 3, 4...

n

n n n i i

i

h

f

x h

n

−

=

−

∑

=

. (3.1)

Then

{ }

h

_n is total over

X

such that

h x

_i

(

_j

)

=

δ

_ij,

i j

,

 

. Therefore, there exists an associated Banach space

1

{{ ( )} :

}

d n

X

=

h x

x



X

equipped with norm

1

{ ( )}

h x

n X_d

=

x

X



 

,

x



X

. Hence

{ }

h

n is a Parseval

d

X

-frame for X. Further,

h

n

∉

[ ]

h

i i n≠ , for all

n

 

. Hence by Lemma 3.1,

{ }

h

n is an exact Parseval

X

d -frame for

X

.

Remark: 3.3 Since

{ }

f

_n is total on

X

if and only if it’s finite linear combinations with rational coefficients are *

w

-dense in

X

* and since the conjugate space of every separable Banach space is

w

*-separable. Therefore, in particular, every separable Banach space

X

has an exact Parseval

X

_d-frame.

Regarding consequence of the above theorem for

X

_d -frames, we prove the following theorem.

Theorem: 3.4 A

X

_d-frame

{ }

f

_n

⊂

X

∗ is an exact if there exists a sequence of non-zero operator

{ }

v

n

⊂

L X X

( ,

)

such that

v

_n∗

( )

f

=

f

,

f



[ ]

f

_{i i}n₌₁ and

v

_n∗

( )

f

=

0

,

f



[ ]

f

_{i i n}∞_{= +}₁.

Proof: Let us assume that above statement is true. Then

v

₁∗

( )

f

₁

=

f

₁. Let

x

₁ be in the range of

v

₁ and

y

₁



X

be such that

x

₁

=

v y

₁

( )

₁ and

f x

₁

( )

₁

=

1

. Then, for

i

=

2, 3,...

, we have

f x

_i

( )

₁



0

. Again, let

x

₂ be in the range of

v

₂



v

₁ and

y

₂



X

be such that

x

₂

=

(

v

₂

−

v

₁

)(

y

₂

)

and

f x

i

(

₂

)

=

0

. Then

f x

1

(

2

)

=

0

. Further for

3, 4,....

i

=

, we have

f x

_i

(

₂

)

=

0

. Continuing like this, let

x

_n be the range of

v

_n

−

v

_n₋₁ and

y

_n



X

be such that 2

(

n n 1

)(

n

)

x

=

v

−

v

−

y

and

f

n

(

x

n

)

=

1

. Then, for

i

≠

n

,

(

i

 

)

, we have

(

)

(

)

0,

(

)

0,

,

i n i n

i n

f y

i

n

f x

i

n

−

=

<



= 

_>



Thus we obtain a sequence

{ }

x

_n

⊂

X

∗ such that

f x

i

(

j

)

=

δ

ij,

i j

,

 

. Therefore,

f

n

∉

[ ]



f

i i n≠ , for all

n

. Hence by Lemma 3.1, { }

f

n is an exact

X

_d-frame for

X

.

Next, we prove that following result, regarding stability of

X

_d-frames.

Theorem: 3.5 If

{ }

f

n

⊂

X

∗ be a

X

d-frames for

X

and let

{

g

n

}

⊂

X

∗ be such that

{

g

n

( )}

x

⊂

X

d, for all

x



X

. Then, {

g

_n

}

is a

X

_d-frames for

X

if and only if there exist a constant

K

>

0

, such that

(

)

{

( )

}

min

{

( )}

, {

( )}

}

d d

d

n n _X n _X n _X

f

−

g

x

≤

K

f

x

g

x

,

x



X

.

(4)

{(

n n

)( )}

X_d

1

g

{

n

( )}

X_d f

B

f

g

x

f

x

A





−

≤

_

+

_







, for all

x

∈

X

. (3.1)

Similarly, we obtain

{(

n n

)( )}

X_d

1

f

{

n

( )}

X_d g

B

f

g

x

g

x

A





−

≤

_

+

_







, for all

x



X

. (3.2)

Choosing

(

1

g

)

f

B A

K

=

+

or

(

1

f

)

g

B A

+

according as

{ {

( )}

, {

( )}

}

d d

n X n X

min



f

x

 

g

x



is

{

( )}

d

n X

f

x



or

{

( )}

d

n X

g

x



.

Conversely, let

C

_f and

D

_f are bounds for the

X

_d -frame { }

f

n in

X

∗. Then, for all

x



X

, we have

2

{

( )}

{(

)( )}

{

( )}

(1

) {

( )}

(1

)( {(

)( )}

{

( )}

)

(1

)

{

( )}

(1

)

d d d d d d d

f X n X

n n X n X

n X

n n X n X

n X

f X

C

x

f

x

f

g

x

g

x

K

g

x

K

f

g

x

f

x

K

f

x

K

D

x

≤

−

+

≤ +

−

+

≤ +

  



 

. Therefore,

(1 )

{

( )}

(1

)

f

d

C

X n X f X

K

x

g

x

K D

x

+

  

≤



≤ +

 

for all

x



X

.

Hence, {

g

n

}

is a

X

_d -frame for

X

with bounds ₍₁ ₎

f

C K

+ and

D

f

(1

+

K

)

.

In the following theorem, we give a sufficient condition for a sequence to be a

X

_d-frame for

X

.

Theorem: 3.6 If

{ }

f

_n

⊂

X

∗ be a

X

_d-frames for

X

with frame bounds

A

and

B

. Let

{

g

_n

}

⊂

X

∗ be such that

{

g

_n

( )}

x

⊂

X

_d, for all

x



X

. If there exists non-negative constants



,



,



and



such that

(i) max{_{, , , }_}

(1 max , , ,

A

α β γ δ α β γ δ

−

<

, where

max{ , , , }

α β γ δ

<

1

.

(ii)

{

(

)

}

{

}

{

} {

}

{

}

2 2 2

( )

2 ( )

( )

d d d d

d

n n n _X n _X n _X n _X

X

f

−

g

x

≤

α

f x

+

β

f x

g x

+

γ

g x

+

δ

x

2_X ,

x



X

.

Then {

g

n

}

is a

X

d -frame for

X

with frame bounds

{ } { }

max , , , (1 ) 1 max , , ,

A α β γ δ A

α β γ δ

− +

+ and

{ } { }

max , , , (1 ) 1 max , , ,

B α β γ δ B

α β γ δ

+ +

− .

Proof: Let

ζ

=

max

{ , , , }

α β γ δ

. Then (ii) may be reproduced as:

{(

)( )}

{ {

( )}

{

( )}

}

d d d

n n X n X n X X

f

−

g

x

≤

ζ

f

x

+

g

x

+

x



 

,

x



X

. (3.3)

Now,

{

( )}

{

( )}

{

)( )}

{

( )}

{ {

( )}

{

( )}

} (using (3.3))

d d d

n X n X n n X

n X n X n X X

g

x

f

x

f

g

x

f

x

ζ

f

x

g

x

≤

+

−

≤

+



(5)

(1

) {

( )}

{(1

)

}

d

n X X

g

x

B

x

ζ

−



≤

+

 

,

x



X

.

Similarly,

(1

+

ζ

) {



g

n

( )}

x



X_d

≥

{(1

−

ζ

)

A

+

ζ

}

 

x

X,

x



X

.

Therefore

{

(1 )

}

{

(1 )

}

1

{

( )}

d 1

A B

X n X X

x

g

x

ζ ζ ζ ζ

ζ ζ

− − + +

+

  

≤



≤

−

 

,

x



X

.

Hence,

{

g

n

}

is a

X

d-frame for

X

with the frame bounds

{ } { }

max , , , (1 ) 1 max , , ,

A α β γ δ A

α β γ δ

− +

+ _and

{ } { }

max , , , (1 ) 1 max , , ,

B α β γ δ B

α β γ δ

+ +

− _.

The next two theorems also gives a sufficient condition for a sequence to be a

X

_d-frame for

X

.

Theorem: 3.7 Let

{ }

f

n

⊂

X

∗ be a

X

d-frame for

X

with frame bounds

A

and

B

and let

{

g

n

}

⊂

X

∗ be a

X

d

-Bessel sequence for X with bounds

K

<

A

, then,

{

f

_n

±

g

_n

}

is a

X

_d -frame for

X

.

Proof: Suppose that

R

_T

,

R

_Q are analysis operators of

X

_d-Bessel sequences

{ }

f

_n ,

{ }

g

_n for

X

, respectively. For any

x

∈

X

, we have

{

(

)

( )

}

( )

d d

n n _X T Q _X

f

±

g

x

=

R x

±

R

x

(

)

{

( )}

{

( )}

,

.

d d

T X Q X

X

R x

R

x

B

K

x

X

≤

+

≤

+

∈

Thus,

{

f

_n

±

g

_n

}

is Bessel sequence for

X

. We also have

{

(

)

( )

}

( )

d d

n n T Q _X

X

f

±

g

x

=

R x

±

R

x

(

)

{

( )}

{

( )}

,

.

d _d

T _X Q _X

X

R x

R

x

A K

x

X

≥

−

≥

−

∈

Hence,

{

f

_n

±

g

_n

}

is a

X

_d-frame for X with respect to

X

_d.

Theorem: 3.8 Let

{ }

f

_n

⊂

X

∗ be a

X

_d-frame for

X

with frame bounds

A

and

B

. Let

{

g

_n

}

⊂

X

∗ be such that

{

g

_n

( )}

x

⊂

X

_d, for all

x



X

and let

{

f

_n

+

g

_n

}

be a

X

_d -Bessel sequence for

X

and with bounds

K

<

A

. Then {

g

_n

}

is a

X

_d-frame for

X

with bounds

A

−

K

and

B

+

K

.

Proof: In the light of the frame inequality for the

X

_d-frame { }

f

n and the fact that

K

is a

X

_d-Bessel bounds for the

d

X

-Bessel sequence

{

f

n

+

g

n

}

, we have

(

)

{

( )}

{(

)( )}

{

( )}

{

( )}

{(

)( )}

(

)

,

d d

d

d d

X n X n n X

n X

n X n n X

X

A

K

x

f

x

f

g

x

g

x

f

x

f

g

x

B

K

x

X

−

≤

−

+

≤

+

≤

+

  



 



(6)

REFERENCES

[1] Casazza P.G., Cristenson O. and Stoeva D.T., Frames expansions in separable Banach spaces, J. Math., Anal. Appl., 307: 2 (2005), 710-723.

[2] Casazza P.G., Han D. and Larson D.R. Frames for Banach spaces, Contemp. Math., 247(1999), 149-182. [3] Christensen O., An itroduction to frames and Riesz bases, Appl. Numer. Harmon. Anal., Birkhuser Boston,

MA-(1992).

[4] Christensen O. and Stoeva D.T.,

p

-Frames in separable Banach spaces, Adv. Comput. Math., 18:2 (2003), 117-126.

[5] Christensen O. and Heil C., Perturbation of Banach frames and atomic decompositions, Math. Nachr, 185 (1997), 33-47.

[6] Duffin R.J. and Schaeffer A.C., A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72(1952), 341-366.

[7] Feichinger H.G. and Gröchenig K., A unified approach to atomic decompositions via integrable group representations, In: Proc. Conf. ``Function Spaces and Applications'', Lecture Notes in Math. 1302, Berlin - Heidelberg - New York, Springer (1988), 52-73.

[8] Gabor D., Theory of communications, J.I.E.E (London), 93 (3) (1946), 429-457.

[9] Gröchenig K., Describing functions: Atomic decompositions versus frames, Monatsh. Math., 112(1991), 1-41. [10] Han D. and Larson D.R., Frames, bases and group representation for Banach space, Memoirs, Amer. Math. Soc.,

147:697 (2000), 1-91.

[11] Jain P.K., Kaushik S.K. and Vashisht L.K., On stability of Banach frames, Bull. Korean Math., 37(112) (2001), 1-41.

[12] Singer I., Bases in Banach spaces-II, Springer, New York, 1970.