On Generalized of Laplace-Weierstrass Transform

On Generalized of Laplace-Weierstrass

Transform

V. N. Mahalle

1

, S.S. Mathurkar

2

, R. D. Taywade

3

Assistant Professor, Department of Mathematics, Bar. R.D.I.K.N.K.D. College, Badnera Railway, Maharashtra, India1

Assistant Professor, Department of Mathematics, Government College of Engineering, Amravati, Maharashtra, India2 Assistant Professor, Departmentof Mathematics, Prof. Ram Meghe Institute of Technology & Research, Badnera,

Amravati, Maharashtra, India3

ABSTRACT: The present paper generalizes Laplace-Weierstrass transform to the space of generalized function by defining testing function space

LW

_a,b

. Laplace-Weierstrass transform is defined by taking

  4

2

y x st

e

  

as a kernel, as it

happens to the most natural pair of transformations. We have proved boundedness theorem called characterization theorem for Laplace-Weierstrass transform. Also representation theorem for Laplace-Weierstrass transform is given which states that every abstract structure with certain properties is isomorphic to a concrete structure

KEYWORDS: Laplace transform, Weierstrass transform, Laplace-Weierstrass transform, Integral transform, Testing function space

I.INTRODUCTION

Integral transformis that it transforms difficult mathematical problems to relatively easy problems. Extensions of some transformations to generalized functions have been done from time to time and their properties have been studied by various Mathematicians. Zemanian [8, 9] extended Laplace and Weierstrass transformations to generalized functions. Mathurkar et.al [2,3] discussed the analyticity of Laplace Weierstrass transform with elementary properties. Pathak [4] developed representation theorem for a class of stieltjes transformable generalized function. Pollard [6] studied representation as a Gaussian integral. The representation theorem states that every abstract structure with certain properties is isomorphic to a concrete structure. Zayed [7] also explained various transform to generalized function. There is much scope in extending double transformation to a certain class of generalized functions. Gudadhe and Gulhane [1] created distributional Laplace-Stieltjes transform.

In the present paper we established the boundedness theorem as well as representation theorem. In section [II], we have defined testing function space. We have given the lemma in section [III]. Section [IV] is devoted the boundedness theorem. Representation theorem is given in section [V]. Lastly conclusions are given in section [VI]. Notation and terminology as per Zemanian.

In this work, define the Laplace-Weierstrass transform

F



s

,

x



of a generalized function

f

directly as the application of

f



t

,

y



to

  4

2

y x st

e

  

i.e.









  4

2

,

,

4

1

,

y x st

e

y

t

f

x

s

F

  

For this purpose we construct a testing function space

LW

_a,b which contains the kernel

  4

2

y x st

e

  

for all

f



t

,

y



in some restricted domain.

II. TESTING FUNCTION SPACE The Testing Function Space

LW

_a,b

b a

LW

_, as the linear space of all complex valued smooth functions





t

,

y



on

0



t





,

0



y





such that for each p, q = 0, 1, 2, - - -



,



sup

2 4



,



,

0 0 ,

, ,

2







 

  

  

y

t

D

D

e

y

t

q

y p t y by at

y t q

p b

a







(2.1)

for some fixed numbers

a

,

b

in R

The space LWa,b is complete and a Frechet space. This topology is generated by the total families of countably

multinorms space given by (2.1).

III. LEMMA

For sufficient condition of boundedness theorem require following lemma

If, on the half plane

{

s

:

a



Re

s

}

and {

x

:

b



Re

x

},

G



s

,

x



is analytic and satisfies

G



s

,

x





K

₁

K

₂



s

,

x



2,

where

K

₁

,

K

₂ are constants and if









 

ds

dx

e

x

s

G

y

t

g

y x st i

i i

i

4

2 '

'

,

4

1

,

  

  

 

 

 







 







,

a





&

'





b

(3.1)

then

g



t

,

y



is a continuous function that does not depend on the choice of



&



'and generates a regular generalized function in

LW

_a*_,b. Moreover

LW



g



t

,

y







G



s

,

x



for

a



Re

s

and

b



Re

x

.

Proof: The fact that

g



t

,

y



does not depend on



&



'follows by Cauchy’s theorem and the bound on

G



s

,

x



. Now from equation (3.1)

















' '

4 '

'

4

_,

_,

_&

4

1

,

2 ' 2

'





















  















  

 

   

 

 

G

i

i

e

d

d

a

b

y

t

g

e

y i t i y

t

(3.2)

In view of the bound on

G



s

,

x



it follows that the last integral converges uniformly for all

t

,

y



0

, which implies the continuity of

g



t

,

y



. In view of equation (3.2) and the bound on

G



s

,

x



it can easily be seen that

g



t

,

y



is a regular generalized function in

LW

_a'_,b . Finally from inversion theorem it also follows that

G



s

,

x





LW



g



t

,

y





for

a



Re

s

and

b



Re

x

_.

IV. CHARACTERISTIC THEOREM FOR LAPLACE-WEIERSTRASS TRANSFORM Theorem:

If

F



s

,

x





LW



f



t

,

y





for,



s

,

x







_f







s

,

x



/



₁



Re

s





₂

,



₁'



Re

x





₂'



,

F



s

,

x



is bounded on any subset



'







s

,

x



/

a



Re

s

&

b



Re

x



and





a



b





' of



according to

F



s

,

x





K

P

 

s

x

for some polynomial P depending on

a

&

b

and

K

is constant.

Proof: Necessary condition

We know that

F



s

,

x



is analytic on



_f by analyticity theorem [2, 3]. By the boundedness property of generalized function, there exist a constant

K

and non negative integers

r

and

r

'.









  4

2

,

,

4

1

,

y x st

e

y

t

f

x

s

F

  





 



















   

  

4 ,

, , 0 0

' 2

'

max

4

y x st q p b a r q

r

p

e

K





 



















   

   

    

 

y y x st q y p t y by at

y t r q

r

p

Sup

e

D

D

e

K

2 2

'

4 2

0 0 0 0

max





 

  4 4 2

0 0

0

2 2

'

.

max

y x y by t a s

y t o q

p

r q

r

p

s

p

x

y

Sup

e

e

K

        

   

 









s

x



K

P

 

s

x

F

,



where

P

 

s

x

depends on

a

and

b

. This proves necessary part.

Sufficient condition

Now to complete the proof of the theorem, we assume that



s

:

a



Re

s







_f and



x

:

b



Re

x







_f . Next assume that

Q



s

,

x



is a polynomial which has no zeros in the half plane

a



Re

s

and

b



Re

x

_{and which} satisfies









_

_

2

2 1

,

,

,

2

x

s

K

K

x

s

Q

e

x

s

F

x



,

a



Re

s

and

b



Re

x

where

K

₁

,

K

₂are constants.

Set











s

x



Q

e

x

s

F

x

s

G

x

,

,

,

2



. Then,









 

ds

dx

e

x

s

G

y

t

g

y x st i

i i

i

4

2 '

'

,

4

1

,

  

  

 

 

 







 







,

Satisfies conditions of above lemma and is a member of

LW

_a*_,b

.

Now

let

f



t

,

y





Q

  

D

g

t

,

y



, where

D

represent the generalized differentiation in

LW

_a*_,b. Then

f



t

,

y



_{is also a member of}

LW

_a*_,b







f

t

y



Q



s

x

 

G

s

x



F



s

x



LW

and

,



,

,



,

for

a



Re

s

and

b



Re

x

.

V. REPRESENTATION THEOREM FOR LAPLACE-WEIERSTRASS TRANSFORM Theorem:

Let

f



t

,

y



be an arbitrary element of

LW

_a*_,b and





t

,

y



be an element of

D

 



, the space of infinitely differentiable functions with compact support on



. Then there exist bounded measurable functions

h

_m,n



t

,

y



defined over



such that,

 

1



,

 

,

,



,

,

1

0 1

0

, 4

2

2



 

 

  





r

m n

n m n y m t y by at n m

y

t

y

t

h

D

D

e

f







where

r

and



are appropriate non-negative integers satisfying

m



r



1

and

n







1

Proof: Let





 0 , , , ,bpq _pq a



be the sequence of seminorms.

Let

f



t

,

y



be an arbitrary element of

LW

_a*_,b and





t

,

y



be an element of

D

 



.Then by the

boundedness property of generalized functions we have for an appropriate constant K and a

non negative integers

r

and



satisfying

p



r

and

q







t

y



K

f

_{ab p q}

q r

p

,

max

,





_{, , ,}





 





t

y



D

D

e

Sup

K

_tp _yq

y by at

y t q

r

p

,

max

2 4

0 0

2





       





t

y



D

D

e

Sup

K

yn

m t q n

p m y by at

y t q

r

p

max

,

max

2 4

0 0 '

2



 

     

    



where K’ is a constant which depends only on m, n and hence p, q



,



(

5

.

1

)

max

,

2 4

0 0 ''

2

y

t

D

D

e

Sup

K

f

_tm _yn

y by at

y t n

r

m







    

   





Now let us set,



,



2 4



,



,

,

(

5

.

2

)

,

2







_

t

y



e

 

t

y

m



r

n



y by at r

Then clearly,



t

y





D

 



r,

,





t

y



e

_r



t

y



y by at

,

,

2 4 _,

2









  

On differentiating above equation with respect to t and y successively we get,



































e

  

t

y

t

y

y

t

D

D

_r y by at y

t

,

,

,

4 2 2 





 













_

































































 























 











  

2

,

,

,

2

,

, , , , 2 4 2 2

y

b

y

t

a

y

y

t

a

t

y

t

y

b

t

y

y

t

e

r r r r y by at    









Let us suppose that in



,







' ' ' '



2

,

t

,

y

/

C

t

D

,

A

y

B

Sup

Sup







_r















Then since,

0

4 2 2



  at by y

e





y t

D

D

































































  

t

y

y

t

y

y

t

a

t

y

t

y

b

y

b

y

t

a

e

r r r r

y by

at

,

,

,

2

2

,

, 2 , , , 4 2 2    







































































  

t

y

y

t

y

y

t

a

t

y

t

A

b

A

b

y

t

a

e

r r r r

y by

at

,

,

,

2

2

,

, 2 , , ' ' , 4 2 2    



































































  

t

y

y

t

y

y

t

t

y

t

y

t

e

K

r r r r

y by

at

,

,

,

,

, 2 , , , 4 2 '' ' 2    









where



















,

,

1

2

,

2

max

' ' '' '

a

A

b

A

b

a

K

Hence by induction we can prove that in



for obvious constant

K

iv, which depends on

a

and

b

then,



,



,



,



(

5

.

3

)

4 2 2



    



n d m c r d y c t y by at iv n y m

t

D

t

y

K

e

D

D

t

y

D





_

Using equation (5.3) in equation (5.1), we get







            



n d m c r d y c t y by at iv y by at y t n r

m

Sup

e

K

e

D

D

t

y

K

f

,

max

2 4 2 4 _,

,

0 0 '' 2 2  







,



(

5

.

4

)

max

_, 0 0

y

t

D

D

Sup

K

_tc d_{y r}

y t n r m v  



    



Now we can write,





_{ }





 

  

   

 

  



t y y t y

t y

t

dy

dt

y

t

D

D

Sup

y

t

Sup

,

,

0 0 0

0







,



_{' '}

(

5

.

5

)

L L y

t

D

t

y

D







Hence from equation (5.1), equation (5.4) becomes,



,



(

5

.

6

)

max

,

_{, ' '}

0 0 11

L L r

n y m t y t n r m vi

y

t

D

D

Sup

K

f

 

 

    



_







Let the product space

L

'



L

' be denoted by

 

L

' 2 _{. We consider the linear one to one mapping}









1 1

,

:

 

 











n r m n

y m

t

D

t

y

D

of

D

 



into

 

2 '

L

. In view of equation (5.6) we see that the linear function





_

,

:

_r,



f



is continuous on



D

 



for the topology induced by

 

2 '

L

. Hence by Hahn Banach theorem, it

can be a continuous linear functional in the whole of

 

2 '

L

. But the dual of

 

2 '

L

is isomorphic with



L

, therefore

there exists



L

- function

h

m,n



m



r



1

,

n







1



_{such that,}











 

 



1 1

,

,

,

,

,

,









n r m

r n y m t n

m

t

y

D

D

t

y

h

f

From equation (5.2) we have,











  

 



11

4 2

,

,

,

,

,

2







n r m

y by at n y m t n

m

t

y

D

D

e

t

y

h

f

Using property of differentiation of a distribution by an infinitely smooth function, we get

 











  

  





1 1

4 2

,

,

,

,

1

,

2







n r m

y by at n

m n y m t n m

y

t

e

y

t

h

D

D

f

Now by using property of multiplication of a distribution by an infinitely smooth function, we get

 



 





 

  





11

, 4

2

_,

_,

_,

1

,

2







n r m

n m n y m t y by at n m

y

t

y

t

h

D

D

e

f

V.CONCLUSION

REFRENCES

[1] Gulhane P. A., Gudadhe A. S., “ Representation theorem for the distributional Laplace-Stieltjes transform”, Science journal of GVISH, Vol. II, pp 29-32, 2005.

[2] MathurkarS. S., Gulhane P. A., “ Elementary properties of Laplace-Weierstrass transform with analytic behaviour”, Proceeding of National Conference on Recent Application on Mathematical Tool in Science and Technology (RAMT-2014), May8-9, 2014.

[3] Mathurkar S. S., Dagwal V. J., Gulhane P. A., “ Analytic behavior of Laplace Weierstrass transform”, International Journal of Mathematical Archive-5(10), pp 243-246, ISSN 2229- 5046, Oct. 2014.

[4] Pathak R .S., “ A representation theorem for a class of Stieltjes transformable generalized Function”, 1974.

[5] Pathak R. S., “ Integral transformation of generalized functions and their applications”, Hordon and Breach Science Publishers, Netherland. [6] Pollard H., “ Representation as a Gaussian integral”, Duke Math. J. Vol. 10, pp. 59-65, 1943.

[7] Zayed A.. I., “ Handbook of function and generalized function transformations”, Mathematical Sciences Reference Series, CRC Press, Boca Raton, FL, 1996.

[8] Zemanian A. H., “ A generalized Weierstrass transformation”, SIAM J. Appl. Math.15, 1088- 1105, 1967.