Inversion Formula For Fourier Jacobi Wavelet Transform

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Inversion Formula For Fourier Jacobi Wavelet

Transform

C. P. Pandey and P. Phukan

Abstract:- In this paper an inversion and Plancherel formula for Fourier-Jacobi wavelet transform are investigated. The Calderon’s reproducing formula for this wavelet transform is also obtained. Some applications associated with Calderon’s reproducing formula for Fourier-Jacobi convolution are given.

Index terms:- Calderon’s reproducing formula; convolution for Fourier-Jacobi operator; Fourier-Jacobi transform; Wavelet transform for Fourier-Jacobi operator.

——————————  ——————————

1.

INTRODUCTION

Integral transform involving special functions as kernels have been used by many authors for the construction of wavelets and wavelet transforms. Pathak and Dixit[10], Pathak and Pandey[8] have constructed Bessel wavelets and Laguerre wavelets by using the theory of Hankel transform and Laguerre transform. Wavelets on finite intervals involving solution of Sturm-Liouville system have been studied by U. Depczynski[11]. Fourier- Jacobi transform is another important transform for the construction of wavelet and wavelet transform. In this paper we describe a new construction of wavelets by using the theory of Fourier- Jacobi transform and inversion and Calderon’s reproducing formula are obtained.

2.

PRILIMINARIES

The generalized Legendre function P_( , )  ( )x defined by equation (2.1), which is given below

/2

(1 ) 1

( , )_{( ) 1, ;1 ; ,}

/2 2 2 2

(1 )( 1)

x x

P x F x R

x

 _{   }

  _{  }

 _{ } _  _         _  where

, ; ; 

F a b c z denotes the Gauss hypergeometric function is a

generalization of the Jacobi polynomial 1, .343p . It reduces

to the Jacobi polynomial _P( , )_{( )x}

n

  _{forn, a non-negative}

integer. Integral transforms with generalized Legendre functions as kernels have been investigated by Braaksma and Meulenbeld[1]. Theory and applications of these transforms can also be found in[3,5,6]. The convolution theory developed by Flensted-Jensen and Koornwinder[5] is basis for the present work.

The following normalized form will be used in the sequel.

_R( , )  _{( )x P}( , )  _{( )/x P}( , )  _{(1),x R}

    (2.2)

Let ch t( ) denote cosh( )t and sh t( ) denote sinh( )t .Then set

_{( )} ( , ) _{( 2 )} 1 ( ) 2

t R ch t

i

 

  _{ }

 (2.3)

Moreover as in [7] we can also express

1 1

( , )_{( ) ( ) ( ), ( );1 ; ( )}2

2 2

t t F i i sh t

 

_ _  _       _

  (2.4)

Also, from [7] we know that ( )t is a solution of the initial value problem





1 _{( ) ( ) ( )} 2 2 _{( )}

( )

d d

t u t u t u t

t dt dt  

_ _{  }

 

   (2.5)

u(0) 1, u(0) 0

where

2 1 2 1

( ) (t et et)(et et)  ,    1 0

       

_22_(sht)21_(cht)21_.

Let _( )t be a Jacobi function of the second kind which is a

solution of (2.5) such that _{( )t e}(i )t_{[1o(1)] as t}

 

   . Thus

- 1-i 1

( ) ( ) , ;1 ;

2

2 2 _{( )}

i i

t t

t e e F i

sht

    

  _



 _{ } 



 _{ }

   _   _

 

. (2.6)

One can show that

_( ) c ( )t _( ) c(- )t _( )t (2.7)

where

( ) 2 ( ) ( 1) - 1 i

2 2

i _i

c

i

  _{ }



    

   

 _{  }

   

_  _{ } _

Let p 

L  be the space of all those functions f on 0, such that

————————————————

• C.P.Pandey, Assistant Professor, Department of Mathematics at North Eastern Regional Institute of Science and Technology, India. E-mail: [email protected]

3345

1/

0

, 1

.

p

p p

f f d p



 

 

 _{ }     







An inner product on _L2 _{ is defined by}

, ( ) g(t) d (t) 0

f g f t 



 

where

   

1/2

(2 )

dt   t dt

(2.8)

For _{f L}1 _{ , the Fourier –Jacobi transform [7] of f is} defined by

 

1

ˆ_{( ) (2 )} 2 _{( ) ( ) ( ) ( ) ( )}

0 0

f   f t_ t t dt f t_ t dt

  

    (2.9)

The inverse of (2.9) is given by

2 2

( ) (2 ) g ( ) ( )| ( )| g ( ) ( ) ( )

n

g t t c d t d

R R

  _    _  

   

   (2.10)

Where _{d }   _21/2_c _ 2_{d .} As in [7] we define the convolution

( * )( ) ( ) ( ) ( , , ) ( ) ( ) 0 0

f g x f t g s K x s t dt d s



   , (2.11)

where





 





1 2 1

2

2 ( 1) _{1 2 3}

(₁,₂,₃) ₂

1

1 2 3 2

cht cht cht K t t t

sht sht sht

   

 

 _{ }  



 

, ; 1;1 2 2

B F      

     

 

with

 

2

2 2

( ₁) ₂ ( ₃) 1

, |_{1 2}| _{3 1 2} 2 ₁cht₂cht₃

cht cht cht

B t t t t t

cht

  

    

(2.12)

and zero otherwise. Then the function K t t t



_{1 2 3}, ,



satisfies the

following properties:

(i) K t t t



_{1 2 3}, ,



is symmetric in all the three variables;

(ii) K t t t



_{1 2 3}, ,



0;

(iii)



_{1 2 3}, ,



(₃) 1 0

K t t t d t





 .

Also it has been shown in [7] that



  

( ) ₁ (₂) (₃) _{1 2 3}, , d ₃ 0

t t t K t t t t

_ _  _ 

(2.13)

Applying (2.10) to (2.13), we have

( ,_{1 2 3}, ) ( ) ₁ (₂) (₃) ( ) 0

K t t t _ t _ t _ t d 

  

(2.14)

For 1 p 2 and 1 1 1

p q

    define the strip

{

D_p    i ℂ : | | (2/p-1) },     1 0

.

From [7]we have following

Lemma 2.1 Let 1 p 2, 1 1 1

p q

    and f Lp( ) Then f( )



is holomorphic in D_p and for all 1D_p,

( ) || || || ||

f  f _p  _q



 . (2.15)

If f L1( ), ( ) fˆ  is continuous in D1 and for all D₁

( ) ₁

f  f



 (2.16)

In [7] if fLp( ) and g Lq , f g L* r  then *

f g_rf _p f_q. (2.17)

Moreover, for _{f g, L}1_{( )  we have}

ˆ _ˆ

( * ) ( )f gf( ) ( ) g (2.18)

For any f L2 ( ) , the following Parseval identity holds for the Fourier-Jacobi transform:

_{2 ˆ} 2  

( ) ( )

0 0

f t dt f  d 

 



  (2.19)

3. FOURIER – JACOBI WAVELET

TRANSFORM

The Fourier – Jacobi translation [5] _y off Lp( ) defined by

   

( ) , ( ) , , ( ),0 ,

0

f x f x y f z D x y z d z x y y

      (3.1)

maps ( )x defined on the positive half of the real axis into

the function f x y( , )defined on the upper half of the positive half plane. _y is also called generalized translation. In

terms of this translation convolution (2.11) can also be expressed as

 

 

   

( * ) 0

f g x _yf y g y dy



  . (3.2)

Using Hölder’s inequality it can be shown that

_{ }

( )

f _p f p

y _L L

 _

  (3.3)

and the mapy_yf is continuous for all

 , [1, )

p

f L  p  .

The dilation 1

D_k is given by

( ) (₁),₁ 0 1

D_k t k t k . (3.4)

Using the above translation and dilation the wavelet

 

, 2 1 x

k k

 is defined as follows

( ) ( ) ( )

, 1

2 1 x 2D1 x 2 k x

k k k k k

     (3.5)

Definition 3.1. Admissible Fourier-Jacobi wavelet.

3346

ˆ ( )2 ( ) 0

C_   d 

 

 , where  ˆ( ) is the

Fourier-Jacobi transform of.

Definition 3.2. Continuous Fourier-Jacobi wavelet transform (FJWT)

For ( )tL2( ) and k_{2 1},k0 we define the continuous Fourier – Jacobi wavelet transform with respect to the Fourier- Jacobi wavelet _,  

2 1t

k k

 as follow





(_{2 1}, ) ( ) _{2 1}, ( ), _, ( ) 2 1

J k k J_f k k  f t _{k k} t

( ) _, ( ) ( ) 2 1 0

f t_{k k} t d t

 

 (3.6)

( ) (₁) ( ) 2

0

f t _k k t dt



 

 

( ) (₁) ( ₂, , ) ( ) 0 0

f t k z K k t z dt dz



   (3.7)

provided the integral is convergent. Since by (2.11) and (2.17) _, ( )

2 1

p L k k

   whenever Lp( ) , by Hölder’s inequality the integral (3.7) is convergent for f Lq( ), 1 1 1

p q



   .

Theorem 3.1. If  is an Fourier-Jacobi wavelet and f is a

bounded integrable function in L1( ) , then the convolution ( * ) f is an Fourier-Jacobi wavelet.

Proof:Since  2

 

2

* ( ) ( ) ( ) ( ) ( )

0 0

f d x _x y f y d y d x

     

 



 

2 2 1_{( )} f 2_{( )}

L L



 



Hence _*f_{( )L}2_{( ) .} Moreover    

2 *

( ) *

0

f

C _f d

     _

   

    2 2 ˆ ˆ

( ) 0

f d

  

  



 

  2 ˆ 2

ˆ ( ) ( )

( )₀

f d

L

    

 



   

Hence *f is a Fourier-Jacobi wavelet.

4.

BASIC PROPERTIES OF FJWT

Theorem 4.1. Let and  be two wavelets and f g, be two functions belongs to L2 ( ) , then

(i) Linearity property:

J_(fg k k)(_{1 2}, )J_( )(f k k_{1 2}, )J_( )(g k k_{1 2}, ) Where  and  are any two scalars.

(ii) Shift property:

(J_f)(x)(k k_{1 2}, ) (J_f)(k k_{1 2}, )

Where  _{is any scalar.}

(iii) Scaling property: If c0 is any scalar, then the Fourier-Jacobi wavelet transform of the scaled

function _{f ( )x} 1_f

 

1

c _c c is

1 2 (J f_c)(k k_{1 2}, ) J f k,k

c c

  _ _

(iv) Symmetry property:

( )(_{1 2}, ) ( )( ) 1, 1 1 2

J f k k J f

k k



 

 _ 

 

 _{ }

 

(v) Parity property:

(J_ppf)(k k_{1 2}, ) (J_f)



k₁,k₂



Where p is the parity operator defined by

( ) ( )

pf xf x .

Proof: The proof is the straight forward application of Fourier-Jacobi transform.

5.

PLANCHEREL

AND

PARSEVALS

FORMULA FOR FOURIER-JACOBI WAVELET

TRANSFORM:

This section describes important properties of the Fourier-Jacobi wavelet transform, such as the Plancherel, inversion formula and associated convolution first, we establish the following theorem.

Theorem 5.1 (Plancherel formula for Fourier-Jacobi

wavelet transform): Let L2 ( ) and ˆ ( )2 ( ) 0, 0

C_   d 

 

  then

for any f g L, 2( ) ; We get

 





 





(₁) (₂)

, _{2 1}, _{2 1},

0 0 1

d k d k

C f g J f k k J g k k

k

 

  



   (5.1)

To prove the above theorem, let us prove the following lemma:

Lemma 5.1: Let L2 ( ) be any basic wavelet then, ˆ _, ( ) ˆ(₁ ) (₂)

2 1 k k

k k

   _{ }

From (3.6) we have





( ) ( ) , , ( )

, 1 2

2 1 x ₀ k z k x z d z

k k

    

(₂) ( ) ( ) (₁ ) ( ) ( ) 0 0

k x z k z d z d

_ _ _    



  

ˆ

(₂) ( ) (₁ ) ( ) 0

k x k d

_ _     

 



ˆ (k₁ ) (k₂)



( )x

3347

Therefore ˆ _, ( ) ˆ(₁ ) (₂)

2 1

k k

k k

   _{ } (5.2)

Proof: We have

( )(_{2 1}, ) ( ) _, ( ) ( ) 2 1 0

J_f k k f x_{k k} x d x



 

( ), _, ( ) 2 1

f x_{k k} x



ˆ( ),ˆ _, ( ) 2 1

f  _{k k} 



ˆ_{( )}ˆ _{( ) ( )} ,

2 1 0

f  _{k k}   d



 

ˆ( ) (ˆ ₁ ) ( ₂) ( ) 0

f   k _ k d 



 





ˆf( ) ( ˆ k₁)





 

k₂

Similarly,

(J g k_ )(_{2 1},k)



gˆ( ) ( ˆk₁)

  

 k₂

Then

(₁) ( ₂) ( )( _{2 1}, )( )(_{2 1}, )

0 0 1

d k d k

J f k k J g k k k

 

 

  



_ˆ







(₁) ( )

ˆ ˆ ˆ

( ) (₁ ) ( ) (₁ )

0 0 1

d k d

f k g k

k

  

     



  

( ) ( )

2 ₁

ˆ

ˆ( ) (₁ ) ˆ( )

0 0 1

d k d

g k g

k

  

   

   

2

ˆ (₁ ) _ˆ

ˆ (₁) ( ) ( ) ( )

0 1 0

k

d k f g d

k

 

    

 

  

Let us take k₁  , then we get 2

ˆ ( ) ( ) _{ˆ ˆ} ( ) ( ) ( )

0 0

d

f g d

    _{   }

 

 

  

ˆ ( )2 ( ) ˆ( ) ( )ˆ ( )

0 0

d f g d

  _{     }



 

  

ˆ_{( ) ( )ˆ ( )} 0

C_ f  g   d



 

C_ f g,

Theorem 5.2 (Inversion formula): If f L2 ( ) , then

 





( ) ( )

1 1 2

( ) _{2 1}, _, ( )

2 1 ₁

d k d k

f x J f k k _{k k} x

C k

 

  

   (5.3)

Where ˆ ( )2 ( ) 0

C_   d 

 

 

Proof: For any g L2 ( ) , we get

  ˆ ˆ

, 2 ( ) ( ) ( )

0

C_ f g_{L } C_ f x g x d x



 

(₁) ( ₂) ( )( _{2 1}, )( )( _{2 1}, )

0 0 1

d k d k

J f k k J g k k k

 

 



  

(₁) (₂) ( )(_{2 1}, ) ( ) _, ( ) ( )

2 1

0 0 0 1

d k d k

J f k k g x _{k k} x d x

k

 

 



 

   

(₁) (₂)

( )(_{2 1}, ) _, ( ) ( ) ( )

2 1

0 0 0 1

d k d k

J f k k _{k k} x g x d x

k

 

 

     

 

(₁) (₂)

( )(_{2 1}, ) _, ( ) ,

2 1

0 0 1

d k d k

J f k k _{k k} x g x

k

 

 



  

Therefore

( ) ( )

1 _{1 2}

( ) ( )(_{2 1}, ) _, ( ) 2 1

0 0 1

d k d k

f x J f k k _{k k} x

C k

 

 

 

  

If fg, then

   2

2 _,

2

2 0 0

dbda

f _L J f b a

a

 



  

Moreover the Fourier- Jacobi wavelet transform is isometry

from _L2 _{ into L}2_{ L}2_{  .}

6.

CALDERON’S FORMULA

In section 3, the wavelet _, ( ) 2 1

x k k

 defined by taking

translation operator first and then dilation operator. In this section to study the boundedness and continuity result we define the wavelet _, ( )

1 2 x

k k

 is defined by taking dilation

operator first and then translation operator. Moreover by using the wavelet _, ( )

1 2 x

k k

 and corresponding Fourier

Jacobi wavelet transform and Fourier Jacobi convolution we obtain Calderon’s reproducing identity and its applications.

Theorem 6.1: Let Lp( )

,

( ) 1

1 _{1 1}

t t

k _{k k}

  _ _

 for k1 0  and

2 ( )

f L then

_{( ) * * ( )} (1)

1 1

0 1

d k

f x _{k k} f x

k



 

 

   

  (6.1)

and we can convert the above expression in the following form

_{( )}

 



_,



2 (2) (1)

2 1 2

0 0 1 ₁

x k d k d k

f x J f k k

k _k

 

 

   

 

   _{ }

  (6.2)

where

 



_{2 1},





₂



( ) ( ) 1

0

J_f k k _k x k f t dt



 

and

2 ( ₂)

1 ₁

x k x k

k _k

  _  _  .

The Fourier-Jacobi wavelet is defined as follows:

Let us take Lp( ) be given, and for k₂0 and k₁0, we get

( ) ( ) ( , )

, 2

1 2 x D1 2 x D1 k x

k k k k k

     

1 2 ,

 

₂, 1

1 1 1

k _x

k x k

k k k 

 

 

 _{ }

 

1 _{( )} 2_{, , ( )}

1 0 1 1

k x

z z d z

k   k k 

  

 

  _{ }

3348

Then the Fourier-Jacobi wavelet transform for the wavelet ( )

, 1 2

x k k

 is defined as follows

J k k(_{1 2}, )

 

J_f (k k_{1 2}, ) ( ), _, ( )

1 2

f x _{k k} x



( ) _, ( ) ( ) 1 2 0

f x_{k k} x d x



 

1 _{( ) ( )} 2_{, , ( ) ( )}

1 0 0 1 1

k _x

f x z z d z d x

k   k k  

  

 

   _{ }

  (6.4)

provided the integral is convergent.

Theorem 6.2: Let f L p( ) and Lq( ) with 1p q,  and 1 1

1

p q  , andJ(Jf)(k k1 2, ) be continuous wavelet

transform(6.4). Then

(1) J k k(_{1 2}, ) is continuous on 0,  0,.

(2)

 





1

, _{, ,} ,

1 2 _,

r

J f k k a f _{p q}

r   

 _ 

1 1 1

1, 1 p q,

r p q

       

_{   }        

(3)

 





1

1

1 1

, , 1

1 2 _, , ,

q

J f k k a f

p  q _{p q}

 _  

 _

  _{   }

 

     

    

Proof:

(1) Let (₂ ,₁ ) 0 0

k k be an arbitrary but fixed point in

(0, ) (0, )   . Then by Hӧlder’s inequality.

(_{1 2}, ) (₁ ,₂ ) 0 0

J k k J k k

2

1 _{2 0}

( ) ( ) , , , , ( ) ( )

0 0 1 1 1₀ 1₀

k

k _{x x}

f x z z z d x d z

a   k k  k k  

  

  _   _{ }

 



   _ _ _ _

 _{ }

 

1

2

1 _{( )} 2_{, ,} 0_{, , ( ) ( )}

0 0 1 1 1₀ 1₀

k p

k _{x x}

p

f x z z d x d z

a  k k  k k  

_{  }   

 _{ }   

 _  _ _ _ _ _

 

 

1

2₀ 2

( ) , , , , ( ) ( )

0 0 1 1 1₀ 1₀

k q

k _{x x}

q

x z z d x d z

k k k k

    

_{  }   

 _{ }   

_  _ _ _ _ _

 

 

Since

2₀

2 , , , , ( ) 2

0 1 1 1₀ 1₀

k

k x x

z z d x

k k k k

  

 

   _{ }

   

 _{ }  

 

  _{ } , by dominated

convergence theorem and continuity of 2 , , 1 1

k x

z k k

_ _

 in

the variable k₂ and k₁, we have

lim (_{1 2}, ) (₁ ,₂ ) 0 0 0 2 2₀

1 1₀

J k k J k k

k k

k k

 

 

This proves that J k k(_{1 2}, )is continuous on (0, ) (0, )   .

(2) From (7) the integral (3.2) is convergent for almost all x, 0 x and

h*_r,h_p,_q,

(3) It can be proved using Hӧlder’s inequality.

Theorem 6.3: If _{f L}1_{( )L}2_{( ) , then f can be reconstructed} by the formula

_{( )} 1 _{( )( , )} 2_, (1) (2)

1 2 2

0 0 1 1 ₁

k x d k d k

l x J l k k

C k k _k

  

 

  

 

   _{ }

 

(6.5)

Where ˆ ( )2 ( ) 0

C_   d 

 

 and(J_f)(k_{2 1},k)is Fourier-Jacobi wavelet transform, we get

(₁) (₂) , ( )(_{1 2}, )( )( _{2 1}, )

0 0 1

d k d k

C f g J f k k J g k k

k

 

  

   

(₁) (₂) ( )(_{1 2}, ) ( ) _, ( ) ( )

1 2

0 0 0 1

d k d k

J f k k g x _{k k} x d x

k

 

 



  

 

   _ _

 

( ) ( )

1 2 1 2

( )(_{1 2}, ) ( ) , ( )

0 0 0 1 1 1 1

k x d k d k

J f k k g x d x

k k k k

 

 



    

   

   _ _{  }

 

 

( ) ( )

2 1 2

( )(_{1 2}, ) , ( ) ( )

2

0 0 0 1 1 ₁

k x d k d k

J f k k g x d x

k k _k

 

 



 

 _   _

 

   _{   }

 _{ } 

 

( ) ( )

2 1 2

( )(_{1 2}, ) , , ( )

2

0 0 1 1 ₁

k x d k d k

J f k k g x

k k _k

 

 

  

 

   _{ }

 

(6.6)

Therefore

( ) ( )

2 1 2

( ) ( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k x d k d k

C f x J f k k

k k _k

  

 

  

 

   _{ }

  (6.7)

If we take fg in equation (6.6) we get

2 _{( )( , )}2 (1) (2)

1 2 2

0 0 ₁

d k d k

C f J f k k

k

 

 



   .

Lemma 6.1: Assume L2 ( ) be a basic Fourier-Jacobi wavelet that satisfies the admissibility condition

ˆ ( )2 ( ) 1 0

C_   d 

 

  . (6.8)

Then for _{f L}1_{( )L}2_{( )}

( ) ( ) ( )

2 1 2 1

( )(_{1 2}, ) , * * ( )

2 _{1 1}

00 1 1 ₁ 0 1

k x d k d k d k

J f k k f _{k k} x

k k _k k

  

  



   _{ }

 

  _{ }  _ _

 

(6.9)

Proof: From equation (6.4) we get ( ) ( )

2 1 2

( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k x d k d k

J f k k

k k _k

  



  

 

  _{ }

 

_{* ( )} 2_, (1) (2) 2 2

1

0 0 1 1 1

d k d k

k x

f _k k

k k k

    

 _   

 _{    }

  _{ } (6.10)

( ) ( )

2 1 2

( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k x d k d k

J f k k

k k _k

  



  

 

  _{ }

 

 

(1) (2)

* (₂) ,₂

1 1

0 0 1

d k d k

f _k k _k x k

k

 

 

 _ 

 _{  }

3349 ( ) ( )

2 1 2

( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k x d k d k

J f k k

k k _k

  



  

 

  _{ }

 

(₁)

* * ( )

1 1

0 1

d k

f _{k k} x

k



 

 _ 

 _{ }

 

Theorem 6.2: Assume that h,L1( ) and (Jh J)()L1( ) holds the admissibility condition

_{( )( )( )( )}ˆ ˆ ( ) ₁ 0

d h     

 



 .

Then the following Calderon’s reproducing identity holds (₁) ₁

( ) * * ( ) , ( )

1 1

0 1

d k

f x f _{k k} x f L

k



  

 

_{  } 

  .

Proof: If we put h in the above lemma, then we can find

the proof of this theorem.

7.

APPLICATIONS

Theorem 7.1: Assume L2 ( ) be a basic Fourier-Jacobi wavelet and

 

J_f



k k_{1 2},



be continuous Fourier-Jacobi wavelet transform, then

( ) ( )

2 1 2

( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k _x d k d k

J f k k

k k _k

  



  

 

  _{ }

 

 ( ) ( ) 0

x

C_ _{ } Jf   d

 _{ }



 _{ }

 

Proof: From equation (5.1), we get

( ) ( )

2 1 2

( )(_{1 2}, ) ,

2

0 0 1 1 ₁

k x d k d k

J f k k

k k _k

  



  

 

  _{ }

 

( ) ( )

2 1 2

( )(_{1 2}, ) ( ) , , ( ) 2

0 0 0 1 1 ₁

k x d k d k

J f k k z k z d z

k k _k

 

 



    

   

   _ _{  }

 

 

(₁) (₂) 2

( )(_{1 2}, ) ( ) ( ) ( ) ( ) ₂

0 0 0 0 1 1 1

d k d k

k x

J f k k z z d d z

k k k

        

   

 _       _

     



   _ _{    } 

       

 

( ) ( )

2 1 2

( )(_{1 2}, ) ( ) ( ) ( ) ( )

2

0 0 0 1 1 0 ₁

k x d k d k

J f k k z z d z d

k k _k

 

      

   

 _      _

 

   

    _{   } _ 

      

 

  ( ) ( )

2 1 2

( )(_{1 2}, ) ( ) ( )

2

0 0 0 1 1 ₁

k _x d k d k

J f k k J d

k k _k

 

     

  

 _     _

   

    _{   } 

     

 

 _{( )} 2 _{( )( , ) ( )} (1) ( )

2 1 2 2

0 0 1 0 1 ₁

k d k d

x

J J f k k d k

k k _k

  

_   _{ } 

   _   _

   

   _{ }  _{ } 

 

     

 _{( ) ( )( , )} (1) ( )

2 1 2

0 0 1 1 ₁

d k d

x

J J J f k k

k k _k

   

_   _

   _{ } 

   

   _{    }

   

Let us take 1

k  , then we get

 _{( ) ( )( , )}  (1) ( )

1 2 1

0 0 1

d k d

x

J k J J f k k

k

  



_{ }   _ 

 _{ }

 

      

 _{( )} _{( )( )( )} (1) ( )

1 1

0 0 1

d k d

x

J k J k J f

k

  



_{ }     _ 

 _{ }

   _ _

  2  (₁) ( )

(₁ ) ( )

0 0 1

d k d

x

J k Jf

k

  



_{ }   

 _{ }

    

 

   

2 (₁ )

( ) (₁) ( )

0 0 1

J k

x

Jf d k d

k

  

_{ }    

 _{ } 

  _ _ 

 ( ) ( ) 0

x

C_ _{ } Jf   d

 _{ }

   

  .

8.

CONCLISION

In this paper we describe a new inversion and Plancherel formula for Fourier-Jacobi wavelet transform. Also we construct a Calderon’s reproducing formula for Fourier-Jacobi wavelet transform. Some applications associated with Calderon’s reproducing formula for Fourier-Jacobi convolution are also given.

REFERENCES

[1] B.L.J. . Braaksma and B. Meulenbeld, Integral transforms with generalised Legendre functions as kernels, Composito Mathematica, 18(1967), 235-287.

[2] C. K. Chui, An Introduction to Wavelets, Academic Press, New York (1992).

[3] J. J. Batancor and B. J. Gonzalez, A convolution operation for a distributional Hankel transformation, Studia Methematica, 117(1) (1985), 57-72.

[4] L. Debnath, Integral transforms and their applications, CRC Press, New York (199

[5] M. Flensted – Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Math, 11(1973), 245-262.

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