Inversion theorem for Laplace-Weierstrass transform

(1)

International Journal of Research in Engineering & Applied Sciences Email:- [email protected], http://www.euroasiapub.org

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75 Inversion theorem for Laplace-Weierstrass transform

V. N. Mahalle1_,

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

S.S. Mathurkar2_,

Assistant Professor, Department of Mathematics, Government College of Engineering, Amravati, Maharashtra, India

R. D. Taywade3

Assistant Professor, Departmentof Mathematics, Prof. Ram Meghe Institute of Technology & Research, Badnera, Amravati, Maharashtra, India

ABSTRACT: We know that for the application of an integral transform, the main condition is the validity of the inversion theorem which allows one to find an unknown function by knowing its image. Keeping this in view, in the present paper we have proved the inversion theorem by proving some lemmas required for the proof of inversion theorem for Laplace-Weierstrass transform. Also uniqueness theorem is proved.

Keywords: Laplace transform, Weierstrass transform, Laplace-Weierstrass transform, testing function space.

I. Introduction:

In mathematics the Laplace and Weierstrass transform are most important and

useful integral transforms. These transforms takes a function of positive real variables t and y to

a function of a complex variable s, x respectively. Laplace-Weierstrass transform are usually

restricted to functions of t and y with t, y > 0. A consequence of this restriction is that the

Laplace-Weierstrass transform of a function is a holomorphic function of the variables s and x. As

a holomorphic function, the Laplace-Weierstrass transform has a power series representation.

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76

Laplace transform takes a function of a complex variable s ( frequency) and yields a function of a

real variable t ( time). Laplace transformation from the time domain to the frequency domain

transforms differential equations into algebraic equations. It has many applications in the

sciences and technology.

Bilodeau [1] proved inversion formula for the Weierstrass transform. Kene and

Gudadhe [2] had presented the some properties of generalized Mellin Whittaker transform.

Mathurkar et.al [3,4] discussed the analyticity of Laplace Weierstrass transform with

elementary properties. Pathak [5] extended integral transform to the compact support. Robbin

and Huang [6] developed inverse filtering for linear shift-variant imaging system. Thakur and

Tamrakar [7] created convergence and inversion theorem for generalized Weierstrass transform.

Zemanian [8] had studied integral transforms like Laplace, Mellin, Hankel, K, Weierstrass in

distributional sense. Motivated by above, in this paper we have formed the inversion theorem for

Laplace-Weierstrass transform.

This paper emphasizes as follows. In section II we defined testing function

space for Laplace-Weierstrass transform. Section III gives three Lemmas required for inversion

theorem In section IV inversion theorem & Uniqueness theorem for the same. And paper is

concluded lastly in section V.

II. The Testing Function Space LWa,b:

b a

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



 

t

,

y

on 0t,

 

 y

(3)

77  

,

sup

2 4

 

,

0 0 ,

, ,

2







 

  



e

D

t

y

t

_tp _yq

y by at

y t q

p b

a





(2.1)

for some fixed numbers

a

,

b

in R

The spaceLWa,bis complete and a Frechet space. This topology is generated by the total families

of countably multinorms space given by (2.1).

The proof of the inversion theorem requires following lemmas.

III. Lemmas:

3.1 Lemma 1:

Let

LW f t y



 

,





F s x

 

,

, for



1



Re

s





2 and

' 2 '

1 Re



 x . Let

 

t

,

y



D

 





, where

 

t

,

y





and set

 

dt dy e

y t x

s

y x st

4

0 0

2 ,

4 1 ,

   











_{. Then for}

any fixed real numbers r and r’_with

₀

  

_r

_{& 0}

  

_r

'

 

_{ }

 

 

 

  

 

  



r

r r

r

h x sg r

r r

r

h x sg

d

x

s

e

h

g

f

d

x

s

e

h

g

f

'

2 '

'

2

,

4

1 ,

,

4

1

₄ ₄

_

_

_











(3.1.1)

Where s



i



_and

x





'



i

 

,

&



'_{are any fixed real numbers such that} 2

1





_and ' 2 '

1





_.

Proof:- The case



 

t

,

y



0

is trivial. Let us consider



 

t

,

y



0

for any

 

t

,

y





. Since

 

s

x

F

,

is analytic for 𝜎1< 𝑅𝑒𝑠 < 𝜎2and 2' '

1 Re



 x _and



 

s

,

x

is an entire function,

then the r. h. s. side integral of the equation (3.1.1) exists. We first show that,

 

, ,

4 1 ,

'

2

4











g h e s x d d

r

r r

r

h x sg

 

 

  

 as a function of

 

g

,

h

which is belongs to

b a

LW _,

(4)

78  

g

h

Sup

e

D

_gp

D

_hq

 

g

h

h bh ag

h g q

p b

a

,

4 2

0 0 ,

, ,

2





 

    



 

 

 

   



    

 r

r r

r

h x sg q

h p g h bh ag

h g

d d x s e

D D e

Sup









,

4

1 '

'

2 2

4 4

2

0 0

Carrying the operator

A

q h p g

D

within the integral and summation sign, which is easily justified

due to smoothness of the integral, we get

 

_{ }

 

 

   



    

 r

r r

r

h x sg q h p g h

bh ag

h g q

p b

a g h Sup e D D e



s x d



d









,

4 1 ,

'

2 2

4 4

2

0 0 ,

, ,

=

  

   





 



     

 

  

  

r

q bh h h x g a s p r

r h

g

d d h x P e

s x s

Sup









4 2

] [

0 0

2 2 '

' , 4

1

(3.1.2)

Where Pq is polynomial in q. The series in right hand side is series of positive finite terms

which is bounded by K for g> 0 and h> 0. Therefore



_a_,_b_,_p_,_q



 

g,h , hence



 

g

,

h

belongs to

b a

LW _, . Therefore r. h. s. of (3.1.1) is meaningful.

Next, partition the path of integration on the straight line from sirtosirinto m intervals,

each of length

m r

2

and from ' ' ir

x



 to x



'ir'_{into n intervals each of length}

n

r

'

2

.

Let si 



iri be any point in the ith interval and

' '

j j

ir

x







be any point in the jth

interval. Consider,

 





  _

     

  

 



_mnrr

e x s h

g

h x g s m

i n

j

j i n

m

j i

' 4

1 1

,

4 ,

4 1 ,

2











  _

     

  

 



_mnrr

e x s

h x g s m

i n

j

j i

' 4

1 1

2 ,

1

2



(5)

79

(3.1.3)

Operate

f

 

g

,

h

to above equation (3.1.3) term by term, we get

 

_





      



m i n j h x g s j i n m

mn

r

e

x

s

h

g

f

h

g

h

g

f

j i 1 1 ' 4 ,

2 ,

1 ,

,

2







 

,

 



,



2 (

3 .

1 .

4 )

1

' 1 1 4 2

mn

r

x

s

e

h

g

f

_i _j

m i n j h x g si j







    



In view of the fact that

 



i j



h x g s

x

s

e

h

g

f

j i

,

1

₄ 2





  

is continuous function on

r

r 





and ' '

r







, the sum on the right hand side of equation (3.1.4) tends to l. h. s. of equation (3.1.1) as m,n.

So we need to show that, for each fixed pandq,



_m_,_n

   

g,h 



g,h converges uniformly to

zero on 0g and 0  h as

m





_and

n





.

Consider,

   



g

h

g

h



Sup

e

D



mn

   

g

h

g

h



q h p g h by ag h o g n m q p b

a

,

4 2 0 , , , , 2















 



    

 

_

_

_{ }

_

_

 





 

) 5 . 1 . 3 ( , 4 1 2 , 1 1 1 4 0 4 2 ' 4 0 4 2 0 0 ' ' 2 2 2 2



 



                           m i n j r r r r h x sg k q q p h bh ag p h x g s j q p i k q j i h bh ag p h g d d x s e h x P s e mn r r e h x P s x s e Sup j i













Where Pq is the polynomial in q and q is finite quantity.

Notice that  

0

0 4 4 2 2 2





      k q h x sg h bh ag

(6)

80

Therefore, given



0

, we can choose G and H so large that for allg G and hH_,

 



  

1

0 0

4 4 2

'

' 2

2

, 3

4 1



   

    

    

  

 



  



r

r r

r k

q

q p k

q

h x sg h bh ag

d d x s h x P s

e









Which is finite quantity. Since

𝑎

( , )t y 0





the right hand side is finite.

Now for all g G and hH, the magnitude of the second term on the right hand side of

equation (3.1.5) is bounded by ₃ _.

Moreover again for g >G and h >H, the magnitude of the first term on the right hand side of

equation (3.1.5 ) is bounded by

 



  

 



 



mn

r

x

s

h

x

P

s

d

x

s

h

x

P

s

m

i n

j k

q

j i j

q p i r

r r

r k

q

q p

'

1 1 0

1

0

2 ,

,

3

'



  

_ _ _



  



















_

_

_

_

(3.1.6)

Now choose 0' ' 0&n

m _{, so large that, for} 0' '

0 and n n

m

m  _{, the above expression (3.1.6) is less}

than

3 2

. Thus, for all g G and h H_and 0' '

0 and n n

m

m  _{, the r. h. s. of equation}

(3.1.5) is less than  i.e.



_a_,_b_,_p_,_q





_m_,_n

   

g,h 



g,h



_{. Moreover, on} 0 gG,0hH

and r



r,r' 



r' _, _the _expression

 

s P



x h



e  

 

s x

e

h x sg k

q

q p h

bh ag

, 4

1 ₄

0 4 2

2 2





  

  





is uniformly continuous function. Therefore, in view of equation (3.1.6), there exists an

' 1 '

1

and

n

m

_{such that for all} mm₁'and nn₁' _,



_a_,_b_,_p_,_q





_m_,_n

   

g,h 



g,h



 _on

H h and G

g  

 0

0 as well. Thus when



'



1 ' 0,

max m m

m and



1,



, 0, max n n

n ,

   



_m_n g h  g h





q p b

a, , ,



, ,



,



on0g  and

0 

h





.

(7)

81

3.2 Lemma 2:

For



 

t

,

y



D

, set

 

  dt dy e y t x s y x st 4 0 0 2 , 4 1 ,    











_{as in lemma (1) then,}

 













h

y

dy

dt

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

d

x

s

e

t g

h y r r r r h x sg











 











 





        

 

2

2 sin

,

1 ,

4

1

' 0 0 4 4 4 ' 2 2 ' ' 2  













Proof: We shall prove the result by justifying the steps in the following manipulations and by

considering compact support of

𝑎

 

t y

,

D





 



 















e

s

x

d

e

t

y

e

dy

dt

d

y x st r r r r h x sg r r r r h x sg















             



 

4

0 0 4 4 2 ' ' 2 ' ' 2

,

4

1

4

1 ,

4

1

 













dy

dt

y

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

d

x

s

e

t g

h y r r r r h x sg











 











 





        

 

2

2 sin

,

1 ,

4

1

' 0 0 4 4 4 ' 2 2 ' ' 2  













3.3 Lemma 3:

Let ' '

2 1 2

1,a ,b ,b , , ,r and r

a



be real numbers with 2

' 1 2

1 a and b b

a 



 



 . Also let

 

t

,

y



D



_.

Then,

 









_h

_y

dy

dt

C

 

g

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

t g

h y

,

2

2 sin

,

1

' 0 0 4 4 ' 2 2













 











 



   

 

_

 



Converges in LW_a_,_b

to



 

g

,

h

as





'

,

r

(8)

82

Proof: Here we have to prove that

 













dy

dt

 

g

h

y

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

t g

h y

,

2

2 sin

,

1

' 0 0 4 4 ' 2 2





 













 











 



   

 

For this we show that

   

,

0

, ,

,bpq

C

g

h



g

h



a





Now consider,

   

g

h

g

h

C

q p b

a, , ,

,



,





 













 

,

0

2

2 sin

,

1

0 0 ' 4 4 4 2 0 0 ' 2 2 2



























 











 





 



       



h

y

dy

dt

g

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

D

e

Sup

t g

h y q h p g h bh ag h g





  (3.3.1)

 













h

y

dy

dt

 

g

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

t g

,

2

2 sin

,

1

0 0 ' '





 

_











 











 



 

   (3.3.2)

Using equation (3.3.2) in equation (3.3.1), we get

   

g

h

g

h

C

q p b

a, , ,

,



,

(9)

83 







    

 

    

 

  

  



_{ }

 



 



 

 



  

 

 

 



1 1 1

1 '

1 1 1

1

1 1 1

1 '

1 1 1

1 4 4 4

2 0

0 _sin _sin

2 , 2

sin sin

2 , 2

1 ' 1

2 2

2

dt dy y

y r e

t rt e

y h g t

dt dy y

y r e

t rt e

y h g t e

D D e

Sup

y t

h y

q h p g h bh ag

h

g _ _

 

 





As A. H. Zemanian [1] pp. 66 and theorem 3.5.1, we can write

   

,

0

, ,

,bpq

C

g

h



g

h



a





. This completes the proof.

IV Theorem:

4.1 Inversion Theorem:

Let

F

 

s

,

x



LW



f

 

t

,

y



_for

f

x

s, 

. Also let

'

r

and

r

_{are real variables. Then in the}

sense of convergence in D’

Where



'

and

are any fixed positive real nos. belonging to _f ,



₁









₂ and '

2 '

1





_..

Proof: We need to show that for



 

t

,

y



D

 



.

 

   











 



 



 

' 4

,

)

2 .

4 .

4 (

,

2

1 ,

,

4

1

' '

2

r

as

h

g

h

g

f

y

t

ds

dx

e

x

s

F

ir

ir ir

ir

y x st











 



From the analyticity of

𝐴

 

s

x

F

,

on

𝐴

f

 and the fact that the

𝐴

 

t

,

y



has compct support

in D(Ω) it follows that the left hand side equation in (4.4.2) is merely a repeated integral with

respect to t, s, y and x and that the above integrand is continuous on the closed bounded domain

 

,   (4.4.1)

4 1 lim

, 4

' ,

2 '

'

' F s x e dxds

y t f

y x st ir

ir ir

ir r

r

  

 

 



 



 

 



(10)

84

of integration. Therefore, the left hand side without the limit notation can be written as,

 

t

y

F

 

s

x

e

 

dx

ds

dy

dt

y x st ir

ir ir

ir











 

 _ 

 

  

 



4

0 0

2 '

'

' '

,

4

1 ,



 







Letting s



i



and

x





'



i



we get,



4

1 

_{ }



 

0 0

,

y

t



F

 

s x e   d d dydt

y x st r

r r

r





4

2 '

' ,

 





Since Φ(t,y) has a compact support and integrand is a continuous function of

𝐴



t

,

y

,



,





, the

order of integration may be changed. These yields



4

1 

_{ }

 

 

r

r r

r

x s F

'

,

 







t y e dydtd d

y x st

4

0 0

2 ,

   







4

1 

'

_{ }

 ₄2

'

,

4

1

sg x h

r

r r

r

e

h

g

f

  

 

 



 







t y e dydtd d

y x st

4

0 0

2 ,

   



 



































 











 





   

 

dy

dt

y

h

y

h

r

y

h

e

g

t

g

t

r

e

y

t

e

h

g

f

t g

h y

2

2 sin

,

1

2

1 ,

,

'

0 0

4

4 '

2 2

 





Because f belongs to LWa,b and in view of lemma 2 & 3, the last expansion tends to

 

g h

 

g h

f ,

2 1 ,

,







= f

   

g,h, g,h

2

1

_





Which complete the proof of the inversion theorem.

4.2 Uniqueness Theorem:

(11)

85

u f 

 is not empty. If

F

 

s

,

x



U

 

s

,

x

on_f _u, then

f

   

t

,

y



u

t

,

y

in the sense of

equality in '

 



D .

Proof: For any

 

t

,

y





and using inversion theorem, we get

     

t

y

u

t

y

t

y



F

 

s

x

U

 

s

x



e

 

dx

ds

 

t

y

f

y x st ir

ir ir

ir r

r

₄

,

1 lim

,

4

,

2 '

'

' '

'







 



  

 

 



 







(4.2.1)

But given that

F

 

s

,

x



U

 

s

,

x

Therefore equation (4.2.1) becomes



0 ,

 

t

,

y



D

 



Hence

f

   

t

,

y



u

t

,

y

in the sense of equality in D'

 

 .

V. Conclusion:

This paper concludes inversion theorem by using some lemmas and uniqueness theorem for Laplace-Weierstrass transform.

References:

[1] Bilodeau G. G. , 1961, An Inverse formula for the Weierstrass transform, Canad. J. Math., 13, pp. 593-601.

[2] Kene R. V. & Gudadhe A. S., 2012, Some properties of generalized Mellin-Whittaker transform, Int. J. Contemp. Math-sciences, vol. 7, No.10, 477-488.

[3] Mathurkar S. S., Dagwal V. J. & Gulhane P. A., 2014, Analytic behaviour of Laplace

Weierstrass transform, International Journal of Mathematical Archive-5(10), 243-246, ISSN 2229-5046.

[4] MathurkarS. S., Gulhane P. A., 2014, Elementary properties of Laplace-Weierstrass Trans

form with analytic behavior, Proceeding of National Conference on Recent Application on Mathematical Tool in Science and Technology (RAMT-2014), May 8-9.

[5] Pathak R. S., Integral transformation of generalized functions and their applications, Hordon and Breach Science Publishers, Netherland.

[6] Robbin G. M. & Huang T. S., 1972, Inverse filtering for linear shift-variant imaging system, proc. IEEE, 60, 862-872.

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86

Applications, ISSN: 2278-9634, Vol.13, pp. 6-12.

[8] Zemanian A. H., 1968 : Generalized Integral Transformation, Interscience Publishers, New York.