Some results on continuous frames for Hilbert Spaces

Int. J. Industrial Mathematics Vol. 2, No. 1 (2010) 37-42

Some Results on Continuous Frames for Hilbert

Spaces

M. Azhini, M. Beheshti

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. Received 15 October 2009; revised 29 Desember 2009; accepted 13 January 2010.

|||||||||||||||||||||||||||||||-Abstract

In this paper some results of continuous frames are discussed. After giving some basic

denitions about these frames, we give some results about the characteristics of continuous

frames in terms of the synthesis operator and its adjoint. Also we discuss the model of

normalized tight continuous frames, and the orthogonal projection which is in relation

with these frames. Moreover the best approximation of the coecients for these frames is

discussed.

Keywords: Continuous frame, Synthesis operator, Best approximation adjoint. MSC 2000: 41A50, 41A58, 42A16, 42C40, 42C15

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1 Introduction

The theory of frames plays an important role in signal processing because of their resilience to quantization (Goyal, Vetterli and Thao [5]), resilience to additive noise, as well as their numerical stability of reconstruction and greater freedom to capture signal characteristics. Also frames have been used in sampling theory to oversampled perfect reconstruction lter banks, system modeling, neural networks and quantum measurements (Eldar and Forney [4]). New application in image processing, robust transmission over the Internet and wireless (Goyal, Kovacevic and Kelner [6]), coding and communication (Strohmer and Health Jr. [8]) was given.

The concept of discrete frames in Hilbert spaces has been introduced by Dun and Schaeer [2] and popularized greatly by Daubechies, Grossmann and Meyer [3]. A discrete frame is a countable family of elements in a separable Hilbert space which allows stable and not necessarily unique decompositions of arbitrary elements in an expansion of frame elements. Later, the concept of coherent states was generalized by Ali, Antoine and Gazeau [1] to families indexed by some locally compact space endowed with a Radon measure and it leads to the notion of continuous frame. Some results about continuous frames were discussed by Rahimi, Najati and Dehghan in [7]. In this paper, we give other results in a dierent approach. The continuous wavelet transformation and short time Fourier transformation are examples of continuous frames.

In this section, we begin with a few preliminaries that will be needed in the next section.

Assume that H is a Hilbert space and (;) is a measure space with positive measure. A mapping

F: !H is called a continuous frame with respect to (;) if F is weakly measurable and there

exist constants A, B such that:

Akfk 2

Z

j<f;F(!)>j 2

d(!)Bkfk 2

; 8f2H: (1.1)

The constants A and B are called the continuous frame bounds. A continuous frame F is called tight if A=B and normalized tight if A=B=1.

The mapping F: !H is called Bessel if the second inequality in (1.1) holds and in this case B

is called the Bessel constant. If F is Bessel, thenT F :

L 2(

;)!H is weakly dened by

<T F

';h>= Z

'(!)<F(!);h>d(!); h2H:

In the next section, we show that the mappingT

F is well dened, linear and bounded and then we

can dene its adjoint by

T F :

H!L 2(

;)

with

(T F

h)(!) =<h;F(!)>; 8!2:

The operatorT

F is called a pre-frame operator or synthesis operator and T

F is called an analysis

operator. We can dene the operatorS F =

T F

T

F and it can be shown that S

F is a positive and

invertible operator. We callS

F the continuous frame operator of F and denote it by

S F

f = Z

<f;F(!)>F(!)d(!):

2 Main Results

Theorem 2.1.

let (;) be a measurable space and F : ! H be an arbitrary function.

F is a Bessel Function with bound B, if and only if the mapping T : L 2(

;) ! H with T(g) =

R

g(!)F(!)d(!) is well dened, linear, bounded and we have kTk p

B .Its adjoint

is given by

T :

H!L 2(

;);T

x=<x;F(:)>; 8x2H:

Proof:

Let F be a Bessel function with bound B, then for allx2H the function F x:

!C

dened by

F x(

!) =<x;F(w)>; 8w2

is integrable andF x

2L 2(

;) such that

kF x

k 2 L 2

B:kxk 2

; 8x2H

which implies thatkF x

k L

2

p

Bkxk. Therefore for allg2L

2, we have g:F

x 2L

1and this function

is integrable. Hence T is well dened. It is clear that T is linear. Moreover:

j<x;Tg>j=j Z

g(!)<x;F(!)>d(!)j

=

Z

jg:F x

jd

p Bkgk

L 2:kxk

Therefore

kTgk=sup kxk=1

j<x;Tg>j p

Bkgk L

That meanskTk p

B:

Conversely, let T be well dened, linear, bounded and we havekTk p

B. We nd its adjoint as

follows;

<T

x;g> L

2=<x;Tg> H=

Z

g(!)<x;F(!)>d(!)

=<<x;F(:)>;g> L

2 :

For allx2H andg2L

2, hence we have T

(

x) =<x;F(:)>and in particular for allx 2H the

function x!<x;F(:)>is- measurable. Now, we have

kT

k=kTk p

B

Therefore

k<x;F(:)>k 2 L 2

Bkxk

2

thus F isp

B - Bessel.

Theorem 2.2.

Let F : ! H be a continuous frame for H with respect to (;) and with the

frame operator S. If we dene the positive square root of S 1 with

S

1=2, then fS

1=2 F(!)g

!2

is a normalized tight continuous frame and for all f 2H we have:

f = Z

<f;S

1=2

F(!)>S 1=2

F(!)d(!):

Proof:

Suppose that the constants A and B are frame bounds, then AIS BI therefore B

1

I S

1

A

1

I. Hence S 1

> 0 and it follows that S

1=2 exists. For showing that fS

1=2 F(!)g

!2is a frame we examine these specications:

1) we consider': !H that is dened by

w!<f;S 1=2

F(w)>; 8f2H:

we have

'(w) =<S 1=2

f;F(w)>

thus'is measurable.

2) For allf 2H we have the following relation

S(f) = Z

<f;F(!)>F(!)d(!)

By substitution ofS 1=2

f we have

S 1=2(

f) = Z

<S 1=2

f;F(!)>F(!)d(!)

=

Z

<f;S 1=2

F(!)>F(!)d(!)

Therefore

S 1=2

S 1=2(

f) =S 1=2(

Z

<f;S 1=2

F(!)>F(!)d(!))

=

Z

<f;S 1=2

F(!)>S 1=2

F(!)d(!)

Thus

f = Z

<f;S 1=2

F(!)>S 1=2

that follows:

kfk 2=

<f;f >= Z

<f;S

1=2

F(!)><S 1=2

F(!);f >d(!)

=

Z

j<f;S 1=2

F(!)>j 2

d(!):

HencefS 1=2

F(!)g

!2is a normalized tight continuous frame.

Proposition 2.1.

LetF;G: !H be two frames for H with respect to(;) and

U :L 2(

)!L 2(

)

is dened by

Ug(:) = Z

<G(:);S 1

F()>g()d()

for g2L

2 and S is the frame operator of F. Then U is well dened linear and bounded operator.

Proof:

SinceS 1

F is a frame, from theorem (2.1) the function g:S 1

F is weakly integrable

on , whereg2L

2 and therefore the function

t=<G(:);S 1

F()>g() is integrable. If C is the

lower frame bound of F and B is the upper frame bound of G then the frameS 1

F has the upper

frame bound 1=C. Now, ifg2L

2since T is bounded then, G is a frame and theorem (2.1) we have:

kUgk 2 L

2 = Z

j

Z

<G(!);S 1

F()>g()d()j 2

d()

=

Z

j<G(!); Z

S

1

F()g()d()>j 2

d()

B k

Z

S 1

F()g()d() k 2 H

B

C kgk

L 2

<1:

Thus G is well dened and bounded. Besides, it is clear that U is linear.

Proposition 2.2.

Let F be a continuous frame for H with respect to (;) for H and T

F be the

pre-frame operator for F. Then the orthogonal projectionP from L 2(

;) ontoR T

is given by:

P(')() = Z

'(!)<S 1

F(!);F()>d(!); 82; 8'2L 2

that S

F is the operator frame for F.

Proof:

From the proposition (2.1) if we considerF = G= P, then the mapping P is well

dened. Now, it is enough to show that for all'2L

2we have:

P(') = (

'; if '2R T

F

0; if '2R ? T F

=N T

F

Let'2R T

F, then we have:

Therefore, from the denition ofP we have,

P(')() = Z

<f;F(!)><S 1

F(!);F()>d(!); 8'2L 2

:

Also we know,

f =S F

S 1 F

f= Z

<f;F(!)>S 1

F(!)d(!); 8f2H:

Hence, we have:

P(')() =<f;F()>='()

thusP(') =';for all'2L 2

:

On the other hand we know that:

N T

F = f'2L

2(

;);<T F

';h>= Z

'(!)<F(!);h>d(w) = 0g:

If'2N

TF, we have Z

'(!)<F(!);F()>d(!) =<T F

';F()>= 0; 82:

Thus

S 1(

Z

'(!)<F(!);F()>d(!)) = 0

and it follows that

Z

'(!)<S 1

F(!);F()>d(!) = 0

that meansP is the orthogonal projection fromL 2 onto

R T

F.

Proposition 2.3.

Assume that F : ! H is a continuous frame with respect to (;). If we

havef = R

g(!)F(!)d(!) forg2L 2(

), then

kgk 2 L

2 = k

e fk

2 L

2+ k

e

f gk

2 L 2:

In other words, if we dene

e

f(!) :=<f;S 1

F(!)>; 8!2;

then, the function e

f is the best approximation coecient for the expansion of the elements of the

continuous frame.

Proof:

Since Z

(g(!) e

f(!))F(!) = 0

thusg e f 2N

T =R

?

T. On the other hand, we have e

f =<f;S 1

F(:)>2R

T. Therefore from

the Pythagorian identity in Hilbert spaceL 2(

) we have

kgk 2 L

2 = kg

e f+

e fk

2 L

2 = kg

e fk

2 L

2+ k

e fk

2 L 2

References

[1] S.T. Ali, J.P Antoine, J.P. Gazeau, Continuous Frames in Hilbert Spaces, Annals of Physics 222 (1993) 1-37.

[2] R.J. Dun and A.C. Schaeer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72( 1952) 341-366.

[3] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.

[4] Y. Eldar, G.D. Forney Jr., Optimal tight frames and quantum measurement, IEEE Trans. Inform. Theory 48 (2002) 599-610.

[5] V.K. Goyal, M. Vetterli, N.T. Thao, Quantized overcomplete expansions in R

n:

Analy-sis,synthesis and algorythems, IEEE Trans. Inform. Theory 44 (1998) 16-31.

[6] V.K. Goyal, J.Kovacevic, J.A. Kelner, Quantized frames expansions with erasures, Appl. Compu. Harmon. Anal. 10 (2001) 203-233.

[7] A. Rahimi,A. Najati, Y.N Dehghan, Continiuous Frames in Hilbert Spaces, Methods of Func-tional Analysis and Topology 12 (2) (2006) 170-182.