Continuous Generalized Hankel Type Wavelet Transformation

Continuous Generalized Hankel-Type Wavelet

Transformation

V. R. Lakshmi Gorty

Department Computer Engineering, SVKM’s NMIMS University, MPSTME, Vile-Parle (W), Mumbai -400056, India *Corresponding Author: [email protected]

Copyright © 2013 Horizon Research Publishing All rights reserved.

Abstract

In this paper, the generalized Hankel-type wavelet transformation is developed. Using the developed theory of generalized Hankel-type convolution, the generalized Hankel-type translation is introduced. Properties of the kernel Dµ ν,

(

x y z, ,

)

are developed in the study.

Using the properties of kernel the generalized Hankel-type wavelet transformation is defined. The existence of the generalized Hankel-type wavelet transformation is given by a theorem. The boundedness and inversion formula for the generalized Hankel-type wavelet transformation is obtained. A basic wavelet which defines continuous generalized Hankel-type wavelet transformation, its admissibility conditions and the wavelet to the function is proved. Examples have been shown to explain the studied continuous generalized Hankel-type wavelet transformation.

AMS: 44A20, 42C40, 46, 33C05.

Keywords

Continuous Generalized Hankel-Type Wavelet Transformation, Generalized Hankel-Type Transformation, Generalized Hankel-Type Convolution

1. Introduction

Malgonde [1] investigated the following generalized Hankel-type transformation

(

)

( )

1

( )

( )

( )

1 1, ,

0

( ) ,

F t F_{µ ν}f t ν t xt Jν _µ xt ν f x dx

∞ −

= =

∫

_ _{ (1)}

( )

J x

µ being the Bessel function of the first kind of order

1/ 2

µ≥ − .

Let L_p

(

0, ,1∞

)

≤ ≤ ∞p , be defined as the space of real measurable function

φ

on

(

0,

∞

)

for which

( )

1/ , ,

0

,1

p p

p

dx

x

x

p

x

µν ν µ ν

φ

₌



_

∞ −

φ



_

_{≤ < ∞}







∫



(2)

( )

0 x

sup

ess

x

µν ν

x

φ

−

φ

∞

=

_{< <∞}

< ∞

. (3)

For each

φ

∈

L

₁

(

0, ,

∞

)

generalized Hankel-type transformation of

φ

is defined by

( )

1

( )

( )

0

ˆ

_x

_t

_{xt J}

ν

_xt

ν

_{( ) , 0}

_{t dt}

_t

_.

µ

φ

₌

ν

− ∞





φ

_{< < ∞}





∫

(4) From [1], it is known that

φ

ˆ

( )

x

is bounded and continuous on

(

0,

∞

)

and

φ

ˆ

( )

x

φ

₁

∞

≤

.

If

f x

( )

is of bounded variation into a neighborhood of the point

x x

=

₀

>

0,

µ

≥ −

1/ 2

and the integral

( )

/2 0

f x x

ν ∞

∫

exists, then the inversion formula in [2] is given by

( )

( )

( )

(

)

(

)

1

0 0 0 1 0 0

0

1

lim

0

0 .

2

R

R

y

x y J

x y F y dy

f x

f x

ν ν

µ

ν

−

→∞

∫









=





+ +

−





(5)

If

f x x

( )

− −ν µ

and

F y y

2

( )

µ ν+ −1 are in

L

1

(

0,

∞

)

, for

( )

(

)

( )

1

( )

( )

( )

1 1, ,

0

( )

,

F t

F

µ ν

f x t

ν

t

xt J

ν µ

xt

ν

f x dx

∞

−





=

=

∫

_

_

(6)

( )

(

)

( )

1

( )

( )

( )

2 2, ,

0

( )

g

,

F t

F

µ ν

g x t

ν

t

xt J

ν µ

xt

ν

x dx

∞

−





=

=

∫

_

_

for

µ

≥ −

1/ 2

,

the following mixed Parseval formula holds for

F

₁-transformation by [2];

( ) ( )

1

( ) ( )

2

0 0

.

f x g x dx

F y F y dy

∞ ∞

=

∫

∫

If

φ

( )

x

and

ψ

( )

x

are in

L

₁

,

then the following Parseval’s formula holds

( ) ( )

( ) ( )

0 0

ˆ

_t

_ˆ

_{t dt}

_x

_{x dy}

_.

φ

ψ

φ

ψ

∞ ∞

=

∫

∫

(7)

To define the generalized Hankel-type Convolution, there is a need to introduce generalized Hankel-type translation.

(

)

1

( )

( )

1

( )

( )

1

( )

( )

,

0

, ,

D

x y z

t

µν ν

t xt J

ν

xt

ν

t yt J

ν

yt

ν

t zt J

ν

zt

ν

dt

µ ν

ν

µ

ν

µ

ν

µ

∞

− − −





−





−





=

_∫

_

_

_

_

_

_

(8)

is defined.

Properties of the kernel Dµ ν,

(

x y z, ,

)

:

Following [3] properties are established: i) For

0

<

x y

,

< ∞

and 0

≤ < ∞

t

, we have

( )

( )

(

)

( ) ( )

1

( ) ( )

( )

1 2

, 0

, ,

t

µν ν

z

zt J

ν

zt

ν

D

x y z dz

xy

xt J

ν

xt

ν

yt J

ν

yt

ν

µ µ ν µ µ

ν

ν

∞

−

+ −





₌





















∫

(9)

Proof:

(

) ( )

3 1

( )

( )

1

( )

( )

1

( )

( )

,

0

, ,

D

x y z

t

µν ν

z zt J

ν

zt

ν

x xt J

ν

xt

ν

y

yt J

ν

yt

ν

dz

µ ν

ν

µ µ µ

∞

− − −







−





−







=

_∫

_

_

_

_

_

_

_

_

( ) ( )

( ) ( )

( )

( )

( )

( ) ( )

( ) ( )

( )

{

}

(

)

{

}

( ) ( )

( ) ( )

( )

1

1 2

0

1 2 1, ,

1

1 2

1, , ,

, ,

.

t

z

xy

xt J

xt

yt J

yt

zt J

zt

dz

t

F

xy

xt J

xt

yt J

yt

F

t

D

x y z

xy

xt J

xt

yt J

yt

ν ν ν ν ν ν

µν ν

µ µ µ

ν ν ν ν

µν ν

µ ν µ µ

ν ν ν ν

µν ν

µ ν µ ν µ µ

ν

ν

ν

ν

∞

− − − −

− − −

−

− +

















=

_

_

_

_

_

_

_

_









=

_

_

_

_









=

_

_

_

_

∫

Applying the inversion formula of generalized Hankel-type transformation to (8)

( )

( )

(

)

( ) ( )

1

( ) ( )

( )

1 2

, 0

, ,

t

µν ν

z

zt J

ν

zt

ν

D

x y z dz

xy

xt J

ν

xt

ν

yt J

ν

yt

ν

µ µ ν µ µ

ν

ν

∞

−

+ −





₌





















∫

(10)

and hence the result. In particular, taking

t

=

0

, gives

ii)

( )

( )

(

)

1

, 0

, ,

1

t

µν ν

z

zt J

ν

zt

ν

D

x y z dz

µ µ ν

ν

∞

+ −





₌





∫

, (11) i.e. for which

x y

,

>

0,

D

µ ν_,

(

x y z

, , belongs to

)

L

1_{0, ,}ν µ

(

0, .

∞

)

iii)

0

<

x y z

, ,

< ∞

,

D

µ ν_,

(

x y z

, ,

)

≥

0

. (12) iv)

D

µ ν,

(

x y z

, ,

)

=

D

µ ν,

(

y x z

, ,

)

=

D

µ ν,

(

z x y

, ,

)

=

...

(13)

( )

(

)

( )

,

(

)

0

,

, ,

,0

,

.

y

T

φ

x

φ

x y

φ

z D

µ ν

x y z dz

x y

∞

=

=

_∫

<

< ∞

The map

y

→

T

_y

φ

is continuous from

(

0,

∞

)

into 0, .

(

∞

)

Let

p q r

, ,

[

1,

)

and 1.

1 1 1

r

p q

∈ ∞

=

+ −

The

generalized Hankel-type convolution of

φ

∈

L

_p

(

0,

∞

)

and

ψ

∈

L

_p

(

0,

∞

)

is defined by

(

)( )

( ) ( )

0

#

x

x y

,

y dy

.

φ ψ

=

∞

_∫

φ

ψ

(14)

In [7] the integral is convergent for almost all

x

,0

< < ∞

x

and

#

_{r p q}

φ ψ

≤

φ ψ

. (15) Moreover,

p

= ∞

,

then

(

φ ψ

#

)( )

x

is defined for all

x

,0

< < ∞

x

and

is continuous.

If

φ ψ

,

∈

L

₁

(

0,

∞

)

, then

(

_{φ ψ}

_#

) ( )

_∧

_t

₌

_{φ ψ}

ˆ

( ) ( )

_t

_ˆ

_t

_{, 0}

_{≤ < ∞}

_t

_.

₍₁₆₎

In this paper, in terms of the aforesaid generalized Hankel-type translation

T

_y and dilation

D

_a defined by

( )

2

(

)

, ,a

,

/ , / ,

D

x y

a

µν ν

x a y a

µ ν

φ

=

− −

φ

(17)

is a continuous generalized Hankel-type wavelet transformation. Its continuity and boundedness properties are established. An inversion formula is obtained.

2. Continuous Generalized Hankel-Type Wavelet Transformation

Let

ψ

∈

L

_p

(

0, ,1

∞

)

≤ < ∞

p

be given. For

b

≥

0 and

a

>

0

define the generalized Hankel-type wavelet transformation

( )

( )

( )

2

(

)

, , , , ,

,

/ , /

b a

x

D T

µ νa y

x

D

µ νa

b x

a

µν ν

b a x a

ψ

₌

ψ

₌

ψ

₌

− −

ψ

(

) ( )

2

, 0

/ , / ,

,

a

µν ν

D

b a x a z

z dz

µ ν

ψ

∞ − −

=

_∫

(18)

the integral being convergent by virtue of (15).

Using the wavelet

ψ

_{b a,}

,

the generalized Hankel-type wavelet transformation is defined.

( )

(

)

( )

( )

( )

1, , 1, , ,

,

,

,

,

_{b a}

H

b a

H

f b a

f t

t

ν µ ν µ ψ

ψ

=

=

( )

,

( )

0

b a

f t

ψ

t dt

∞

=

_∫

(19)

( ) ( )

(

)

2

, 0 0

/ , / ,

a

µν ν

f t

z D

b a t a z dzdt

µ ν

ψ

∞ ∞ − −

=

∫∫

(20)

provided the integral is convergent. The continuity of the generalized Hankel-type wavelet follows from the boundedness property of the generalized Hankel-type translation [5].

(

0,

)

p

L

∞

. The function

ψ

b a, is defined almost everywhere on

[

0,

∞

)

, and

( )

(2 )(1/ 1)

, p

.

b a

x

_p

a

_p

µν ν

ψ

_≤

+ −

ψ

(21) The existence of the generalized Hankel-type wavelet transformation is given by the following theorem.

Theorem 2. Let

f L

∈

_p

(

0,

∞

)

and 0,

ψ

∈

L

_p

(

∞

)

with

1

≤

p q

,

< ∞

and

1 1 1

p q

+ =

;

( )

(

)

( )

1, ,

,

1, , ,

,

H

ν µ

b a

=

H

ν µ ψ

f b a

be the continuous wavelet transform (19 ). Then

(

H

1, ,ν µ

f b a

)

( )

,

is continuous on

(

0,

∞ ×

) (

0, ,

∞

)

(

)

( )

(

)

( )

2

1, , ,

,

,

_r p q

,

1 1 1

1, 1

, ,

,

H

f b a f b a

a

f

p q r

r

p q

µν ν

ν µ ψ

≤

+

ψ

=

+ −

≤

< ∞

(

)

( )

(

H

1, , ,

f b a f b a

,

)

( )

,

a

(2 ) 1q 1

f

p q

,

1 1

_{p q}

1.

µν ν

ν µ ψ

ψ

  +  −  

∞

≤

+ =

Proof.

Let

(

b a

₀

,

₀

)

be an arbitrary but fixed point in

(

0,

∞ ×

) (

0, .

∞

)

Then by Hölder’s inequality,

(

)

( )

(

)

(

)

( ) ( )

(

)

(

)

( )

(

)

(

)

( )

(

)

(

)

1, , 1, , 0 0

2

, , 0 0 0

0 0

1/ 2

, , 0 0 0

0 0

1/

, , 0 0 0

0 0

,

,

/ , / ,

/ , / ,

/ , / ,

/ , / ,

/ , / ,

/ , / ,

p p

q q

H

f b a

H

f b a

a

f t

z D

b a t a z

D

b a t a z dtdz

a

f t

D

b a t a z

D

b a t a z dtdz

z

D

b a t a z

D

b a t a z dtdz

ν µ ν µ

µν ν

µ ν µ ν

µν ν

µ ν µ ν

µ ν µ ν

ψ

ψ

∞ ∞ − −

∞ ∞ − −

∞ ∞

−





≤

_

−

_





≤

_

−

_









×

_

−

_





∫∫

∫∫

∫∫

Since by (9) _,

(

)

_,

(

_{0 0 0}

)

0

/ , / ,

/ , / ,

2

D

µ ν

b a t a z

D

µ ν

b a t a z dt

∞

−

≤

∫

, by dominated convergence theorem and

continuity of

D

µ ν_,

(

b a t a z

/ , / ,

)

in the variables

b

and ,

a

we have

(

)

( )

(

)

(

)

0 0

1, , 1, , 0 0

lim

,

,

0

b b a a

H

ν µ

f b a

H

ν µ

f b a

→ →

−

=

.

This proves that

H b a

( )

,

is continuous on

(

0,

∞ ×

) (

0, .

∞

)

Inequality (15) proves

(

)

( )

(

)

( )

2

1, , ,

,

,

_{p q}

,

1 1 1

1, 1

, ,

.

r

H

f b a f b a

a

f

p q r

r

p q

µν ν

ν µ ψ

≤

+

ψ

=

+ −

≤

< ∞

It can be proved using Hölder’s inequality.

3. An inversion Formula

In this section, it is shown that the function

f

can be recovered from its wavelet transform when the wavelet

ψ

satisfies admissibility condition.

Theorem 3. Let

ψ

∈

L

2

( )

R

+ be a basic wavelet which defines generalized Hankel-type wavelet transformation (18).

Then, for

( )

2 2

0

ˆ

0,

A

w

µν ν

w dw

ψ

ψ

∞ − −

we have

(

)

( )

(

)

( )

(

(

)

( )

)

( )

2

1, , , 1, , ,

0 0

,

,

,

,

,

H

f b a f b a

H

f b a g b a a

µν ν

dadb A f g

ν µ ψ ν µ ψ ψ

∞ ∞

− −

₌

∫∫

(23)

( )

2

for all ,

f g L

∈

R .

+

Proof. In view of representation (8) we have

(

)

( )

( ) ( )

(

)

1, , ,

, 0 0

,

/ , / ,

H

f b a

f t

z D

b a t a z dzdt

ν µ ψ

µ ν

ψ

∞ ∞

=

∫∫

(

) ( )

( )

( )

{

}

(

) ( )

( ) ( )

( )

1 2

0 0 ₁

1 2

0

1 0

ˆ /

ˆ

_/

_ˆ

ˆ

_ˆ

b

xb

_J

xb

a

a

a

a

f x a

z

dxdz

z zax J

zax

b

xb

xb

a

f x a

x

J

dx

a

a

a

f u

au

b bu

ν ν

µ µν ν

ν ν

µ

ν ν

µν ν

µ

ν

ν

ψ

ν

ψ

ν

ψ

ν

− ∞ ∞

− −

−

− ∞

− −

∞

−





_

_{  }

_



_

_





_







  

_{  }



_





_



_























=

_





_



























  









=



  

_{  }



_





_



_





















=

∫∫

∫

∫

{

( )

}

( ) ( )

(

ˆ

_ˆ

)

( )

_.

J

bu

du

f u

au

b

ν µ

ψ









=

∧

(24)

Parseval identity (7) yields

(

)

( )

(

)

( )

( ) ( )

(

)

( ) ( ) ( )

(

)

( )

( ) ( )

(

)

(

( ) ( )

)

1, , , 1, , ,

0 0 0

,

,

ˆ

_ˆ

_ˆ

_ˆ

ˆ

_ˆ

_ˆ

_ˆ

_.

H

f b a H

f b a db

f u

au

b g u

au

b db

f u

au g u

au du

ν µ ψ ν µ ψ

ψ

ψ

ψ

ψ

∞

∞

∞

=

∧

∧

=

∫

∫

∫

(

)

( )

(

)

( )

( ) ( )

(

)

(

( ) ( )

)

( ) ( )

( ) ( )

( ) ( )

( )

( ) ( )

( )

2

1, , , 1, , ,

0 0

2 0 0

2

0 0

2 ₂

0 0

2 ₂

0 0

,

,

ˆ

_ˆ

_ˆ

_ˆ

_.

ˆ

_ˆ

_ˆ

_ˆ

_.

ˆ

_ˆ

_ˆ

_.

ˆ

_ˆ

_ˆ

_.

, .

H

f b a H

f b a db a

da

f u

au g u

au du a

da

f u g u du

au

au a

da

f u g u du

au

a

da

f u g u du

w

w

dw

A f g

µν ν

ν µ ψ ν µ ψ

µν ν

µν ν

µν ν

µν ν

ψ

ψ

ψ

ψ

ψ

ψ

ψ

∞ ∞

− −

∞ ∞

− −

∞ ∞

− −

∞ ∞

− −

∞ ∞

− −

=

=

=

=

=

∫∫

∫∫

∫

∫

∫

∫

∫

∫

Notice that admissibility condition (22) requires that

ψ

ˆ 0 0

( )

=

. If

ψ

ˆ

is continuous then from (3) it follows that

( )

0

0.

x dx

ψ

∞

=

∫

This justifies the wavelet to the function.

( )

( )

( )

(

)

(

)

( )

( )

( )

( )

( )

( )

, 0

1 1 1

,

0

Now considering

,

,

by putting

and

. Then

,

/ , / ,

.

Since

, ,

.

b t

_{x y}

b

_x

t

_y

_{x y}

_{z D}

_{b a t a z dz}

a a

a

a

D

x y z

x x

J

x

y

y

J

y

z z

J

z

d

µ ν

ν ν ν ν ν ν

µν ν

µ ν µ µ µ

ψ

ψ

ψ

ψ

ξ

ν

ξ

ξ

ν

ξ

ξ

ν

ξ

ξ

ξ

∞

∞

− − − − −



_{ =}

₌

₌

₌

























=

_

_

_

_

_

_

_

_

∫

∫

( )

( )

1

( )

( )

1

( )

( )

1

( )

( )

0 0

Substituting,

,

x y

z

µν ν

x x

ν

J

x

ν

y

y

ν

J

y

ν

z z

ν

J

z

ν

d

dz

µ µ µ

ψ

₌

∞

ψ



∞

ξ

− −



ν

−

ξ



ξ



ν

−

ξ



ξ



ν

−

ξ



ξ



ξ







_

_

_

_

_

_

_

_







∫

∫

( )

( )

( )

{

( )

( )

( )

( )

}

( )

( )

( )

( )

{

}

(

)

( )

1 1 1

0 0

1 1

1, , 0

.

z

z z

J

z

dz

x x

J

x

y y

J

y

d

x x

J

x

y y

J

y

F

d

ν ν _{µν ν} ν ν ν ν

µ µ µ

ν ν ν ν

µν ν

µ µ µ ν

ψ

ν

ξ

ξ

ξ

ν

ξ

ξ

ν

ξ

ξ

ξ

ξ

ν

ξ

ξ

ν

ξ

ξ

ψ ξ ξ

∞ ∞

− − − − −

∞

− − − −



_

_



_

_

_

_

= 

_

_



_

_

_

_













=

_

_

_

_

∫ ∫

∫

(

)

( )

1 1

1, , 0

Substituting

and

.

,

.

b

_x

t

_y

a

a

b t

b

b

_J

b

t

t

_J

t

_F

_d

a a

a

a

a

a

a

a

ν ν ν ν

µν ν

µ µ µ ν

ξ

ξ

ξ

ξ

ψ

ξ

ν

ν

ψ ξ ξ

− −

∞ − −

=

=





















₌

  



_





_

  



_





_









  







  







(

)

( )

( )

(

)

( )

( )

1, , , 2 0 1 2 1, , 1 0 0

,

,

.

H

f b a

b t

a

f t dt

a a

b

b

_J

b

a

a

a

a

F

d

f t dt

t

t

_J

t

a

a

a

ν µ ψ

µν ν

ν ν

µ µν ν µν ν

µ ν ν ν µ

ψ

ξ

ξ

ν

ξ

ψ ξ ξ

ξ

ξ

ν

∞ − − − ∞ ∞ − − − − −





=

_

_









_{  }

_



_

_











_{  }

_



_

_











  





_







_





=









×









  



















  













_

_{  }

_

_

_

_

_

_

_

_







∫

∫ ∫

( )

(

)

( )

(

)

1 1 2 0 0 1, , 1 2 1, , 0

b

b

b

t

t

t

a

J

J

f t dt

a

a

a

a

a

a

F

d

b

b

b

a

J

F

f

a

a

a

a

ν ν ν ν

µν ν µ µ µν ν µ ν ν ν µν ν

µ µ ν

ξ

ξ

ξ

ξ

νβ

ν

ξ

ψ ξ ξ

ξ

ξ

ξ

νβ

ξ

− − ∞ ∞ − − − − − ∞ − − −













  







  







=

  













_

  













_

  





_





_



_

  





_







_

_

×





  







 

=

  

_{  }



_





_



_



 

_{ }

×









∫

∫

∫

µν ν−

(

F

1, ,µ ν

ψ ξ ξ

)

( )

d

.

By substituting

x d

;

adx

,

a

ξ

₌

_ξ

₌

the continuous generalized Hankel-type wavelet transform can be written as

(

)

( )

(

)

(

)

( )

( )

( )

(

)

( ) ( )

(

)

( )

( )

( )

1 2

1, , , 1, , 1, ,

0

1 2

1, , 1, ,

0

2 3/2 1

0

,

b

b

b

.

H

f b a

a

J

F

f

F

d

a

a

a

a

b

a

bx J

bx

F

f x

ax

F

ax adx

a

a

a

b bx J

bx

F

ν ν

µν ν µν ν

ν µ ψ µ µ ν µ ν

ν ν µν ν

µν ν

µ µ ν µ ν

ν ν

µν ν µν ν

µ

ξ

ξ

ξ

ν

ξ

ψ ξ ξ

ν

ψ

ν

− ∞ − − − − − ∞ − − − − ∞ − − − − + −





  







 

=

_{  }

_



_

_



_{ }

×

  





_







_

 

 

_

_

=

_{ }

_

_

×

 





=

_

_

∫

∫

∫

(

)

( )

(

)

( )

( )

( )

(

)

( )

(

)

( )

( )

( )

(

)

( )

(

)

( )

1, , 1, ,

3

2 2 1

1, , 1, ,

0 3

3 2 _{2 1}

1, , 1, ,

0

.

.

f x x

F

ax dx

a

a

b bx J

bx

F

f x x

F

ax dx

a

b bx J

bx

F

f x x

F

ax dx

µν ν

µ ν µ ν

µν ν _{ν ν}

µν ν µν ν

µ µ ν µ ν

µν ν _{ν ν µν ν}

µ µ ν µ ν

ψ

ν

ψ

ν

ψ

− − ∞ − − + − − − − − ∞ − − + _{− − −}





=

_

_





=

_

_

∫

∫

Example 1. Assume that

f t

( )

=

t

−τ

;0

< <

τ

(

2

µ

+

1 .

)

ν

Then the generalized Hankel-type wavelet transforms of

t

−τ is given by

( )

(

)

( )

( )

(

)

1, , , 2 0 2 , 0 0

,

,

/ , / ,

H

t

b a

b t

a

t

dt

a a

a

z

t D

b a t a z dt dz

τ ν µ ψ

µν ν τ

µν ν τ

µ ν

ψ

ψ

− ∞ − − − ∞ ∞ − − −





=

_

_









=

_

_





∫

∫

∫

(25)

(

)

1

( )

( )

1

( )

( )

1

( )

( )

,

0

Since

, ,

D

x y z

µν ν

x x

ν

J

x

ν

y y

ν

J

y

ν

z z

ν

J

z

ν

d

µ ν

ξ

ν

ξ

µ

ξ

ν

ξ

µ

ξ

ν

ξ

µ

ξ

ξ

∞

− −



−





−





−







(

)

( )

( )

1 1

,

0 ₁

;

;

/ , / ,

b

t

x

y

a

a

b

b

_J

b

t

t

_J

t

a

a

a

a

a

a

D

b a t a z

z z

J

z

d

ν ν ν ν

µ µ

µν ν µ ν

ν ν

µ

ν

ξ

ξ

ν

ξ

ξ

ξ

ν

ξ

ξ

ξ

− −

∞ − −

−

=

=



_{  }

_



_

_



_{  }

_



_

_







_{  }

  



_





_



_



_{  }

  

_







_



_

















=



















_

_







∫

(

) ( )

( )

( )

( )

( )

(

) ( )

( )

( )

( )

( )

1 1

,

0 ₁

1 1

,

;

.

/ , / ,

/ , / ,

ax d

adx

b

b

_{ax J}

b

_ax

t

_{tx J}

_tx

a

a

a

a

D

b a t a z

ax

z zax J

zax

adx

b

_{bx J}

_bx

t

_{tx J}

_tx

a

a

D

b a t a z

ax

ν ν

ν ν

µ µ

µν ν µ ν

ν ν

µ

ν ν ν ν

µ µ

µν ν µ ν

ξ

ξ

ν

ν

ν

ν

ν

− −

∞

− −

−

− −

− −

=

=



_{  }

_



_

_



_{ }









_{  }

_



_

_



_{ }

_

_



  











 





=















_

_







 

_

_

 

_

 

_

_

 

 

 

=

∫

( )

( )

0

_{z zax J}

1 ν

_zax

ν

_adx

µ

ν

∞

−









_

_









_

_















∫

Using the representation (8), the inner integral can be evaluated by means of the formula [3, p. 390 (4)]. Thus

(

)

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

, 0

1 1

1

0 0

1

4 2 3 1

0 0

/ , / ,

t D

b a t a z dt

b

t

t

ax

bx J

bx

tx J

tx

z zax J

zax

adx dt

a

a

a

x

zab

J

bx

J

zax

t

J

tx

dt dx

τ µ ν

µν ν ν ν ν ν ν ν

τ

µ µ µ

ν ν ν ν

µν ν µν ν τ ν

µ µ µ

ν

ν

ν

ν

∞

−

− −

∞ ∞

− −

− −

∞ ∞

− +

− − + − + − − +



_{ }

_{ }















=

_

_{ }

_

_

_{ }

_

_

_

_

_

 

 

























=

_

_

_

_

_

_

_

_





∫

∫ ∫

∫

∫

( )

( )

( )

( )

( )

( )

( )

2 1

4 2 3

0

1

3 4

0

1

2

2

1

2

1

2

2

_.

1

2

x

x

a

x

zab

J

bx

J

zax

dx

zab

a

x

J

bx

J

zax dx

τ τ ν

ν ν

ν ν

ν ν ν

µν ν µν ν

µ µ

τ

ν _{µν ν τ ν µν} ν ν

ν

µ µ

µν τ ν

ν

ν

µν τ ν

ν

ν

µν τ ν ν

ν

µν τ ν

ν

ν

− +

− ∞

− + − − + − +

∞ −

− + − + − + −



_Γ



− +









_



_









 



=

_

_

_

_

+ +



_Γ











_

_







− +





_

_

_

_

Γ

_



_

_

_

_

_

=

+ +





Γ

_



_

∫

∫

( )

(

)

( )

(

)

( ) ( )

[

]

( ) ( )

1 2

1, , ,

1 _{5 1 4 1/2}

2 2

2 1

1

1

1

2

2

2

,

1

1

1

₁

2

2

1

1 1

_,

1

_{, 1}

_,

2

2

H

t

b a

a

zab

za

b

a

F

za

b

µν ν

τ µν ν

ν µ ψ

ν µν _{ν τ µν ν}

ν ν

µν τ ν

µν τ ν

ν

ν

µν τ ν

µν τ ν

_µ

ν

ν

ν

µν τ ν

µν τ ν

_µ

_ψ

ν

ν

− +

− +

− + − + − + − −

−



_Γ



− +

 

_Γ

+ + +









 







 







=

×

+ +

−

+ − +



_Γ



 

_{Γ −}



_{Γ +}





 





_

_{ }

_







×





+ + +

−

+ + +



₊







_



_







( )

0

z dz

∞

∫

for

0

< <

τ µν ν

+

, ,

a b

>

0.

Example 2. Assume that

f t

( )

=

t

−3

sin

( )

w t w

₀

;

₀

>

0.

Then,

( )

(

)

(

)

( )

( )

(

)

( )

(

( )

)

(

)

3

1, , , 0

2 3 0 0 2 3 0 , 0 0

sin

,

sin

,

sin

/ , / ,

.

H

t

w t

b a

b t

a

t

w t

dt

a a

a

z

t

w t D

b a t a z dt dz

ν µ ψ

µν ν µν ν µ ν

ψ

ψ

− ∞ − − − ∞ ∞ − − −





=

_

_









=

_

_





∫

∫

∫

Using the representation (8), the inner integral can be evaluated by means of the formula [3, p. 405 (2)]. Thus

( )

(

)

( ) ( )

( )

( )

( )

( )

( )

( )

3 0 , 0 1 1 3 0

0 0 1

sin

/ , / ,

sin

t

w t D

b a t a z dt

b

_{bx J}

_bx

t

_{tx J}

_tx

a

a

t

w t

ax

dt

z zax J

zax

adx

µ ν

ν ν ν ν

µ µ µν ν ν ν µ

ν

ν

ν

∞ − − − ∞ ∞ − − − −



_{ }

_

_

_{ }

_

_





_{ }

_

_

_{ }

_

_



 

 

=







_

_















∫

∫

∫

( )

( )

( )

( )

( )

( )

( )

( )

1

4 2 3 4

0

0 0

4

4 2

1

4 2 3

0 4 4

sin

1

4

2

2

1

4

2

1

2

2

a

x

zab J

bx

J

zax

w t t

J

tx

dt dx

x x

w

a

x

zab

J

bx

J

zax

dx

x w

ν ν ν

µν ν µν ν ν

µ µ µ

ν ν ν ν

ν ν ν

µν ν µν ν

µ µ ν

ν

µν

ν

ν

ν

µν

ν

ν

ν

∞ ∞ − − − + − + − + − + − + ∞ − + − − + − + −

















=

_

_

_

_

_

_

_

_







_Γ



− +









_



_









 



=

_

_

_

_

+ +









Γ





_

_







Γ

=

∫

∫

∫

( )

1

( )

( )

3 4 0

4

.

1

4

2

zab

ν

a

µν ν

x

ν µν

J

bx

ν

J

zax

ν

dx

µ µ

µν

ν

ν

_ν

µν

ν

ν

ν

∞ − + − + − −





− +









_



_























+ +



_Γ











_

_







∫

(

)

(

)

( )

(

)

( )

[

]

( ) ( )

5 6 4

3 15/2

1, , , 0

2 2

4 7 1

2 2

2 1

1

4

1

1

1

2

2

2

2

sin(

,

1

4

1

1

₁

2

2

1

1 1

_,

1

_{, 1}

_,

2

2

x w

H

t

w t b a

a

za

b

F

za

b

ν µν ν

ν µ ψ

µν ν _{ν τ}

ν ν

µν

ν

µν τ ν

µν ν

ν

ν

ν

µν

ν

µν ν

ν

µ

ν

ν

ν

µν ν

µν ν

_µ

ν

ν

− − +

−

− + − −

−



_Γ



− +

 

_Γ

− +

 

_Γ

+ +









 

 







 

 







=

×

+ +

− −



_Γ





_Γ





_{Γ +}













_

_

_

_







×





+ +

−

+ +



₊









( )

0

.

z dz

ψ

∞

_









∫

4. Conclusion

The developed theory of the generalized Hankel-type wavelet transformation can be applied in signal processing, computer vision, seismology, turbulence, computer graphics, image processing, digital communication, approximation theory, numerical analysis and statistics. Generalized Hankel-type wavelet transformation can be used for complex information such as speech, images etc. to be decomposed into elementary forms.

REFERENCES

[1] S. P. Malgonde, Real Inversion Formula for the distributional generalized Hankel type integral transformation, Rev. Acad. Canaria Cienc, XVIII, (2007) (Spain).

[2] Kilbas, Anatoly A. , Trujillo, Juan J. , May 2003, “Hankel-Schwartz and Hankel-Clifford transforms on

L

ν_,p

spaces”, Results in Mathematics, 43, Issue 3-4, pp 284-299. [3] Watson, G. N., 1958, A treatise on the theory of Bessel

functions, Cambridge University Press, London.

[4] S. P. Malgonde G. S. Gaikawad; Nov.2001, On a generalized Hankel type convolution of generalized functions, Proc. Indian Acad. Sci. (Math.sci.), Vol.111, No.4, , pp.471-487. [5] A. H. Zemanian, 1968, Generalized Integral Transformations;

Inter Science Publishers, New York.