International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 618

ISSN 2229-5518

Laplace transform associated with the Weierstrass transform

Dr. P.A. Gulhane and S.S. Mathurkar
Dept. Of Mathematics, Govt. College of Engineering, Amravati, (M.S), India gulhane.padmaja@gcoea.ac , ssmathgcoea@gmail.com

Abstract:

An elegant expression is obtained for the Laplace-Weierstrass transform LW of the function in terms of their existence. It is also shown , how the main result can be extended to hold for the LW transform of several functions.

Keywords: Transform,Laplace Transform,Weierstrass Transform, Laplace-Weierstrass transform, Generalized Function, Existence theorem,Multiplication Theorem.

1. Introduction:IJSER

In this paper we have introduced the
concept of LW transform which has use in several fields. The basic idea behind any

F ( x ) =

1 ∫ f ( y ) e−( x − y ) /4 dy

−∞

(1.2)

transform is that the given problem can be
solved more readily in the transform domain.
where

f (y ) is a suitably restricted conventional

The method is especially attractive in the linear mathematical models for physical systems such
function on
variable.

− ∞ < y < ∞

and x is a complex
as a spring/mass system or a series electrical circuit which involve discontinuous functions.
The distributional Laplace transform is
defined as
Our purpose in this work is to define
and study the Laplace transform associated with the Weierstrass transform.
This transform is defined by

F ( s ) =

f (t ) , e− st

(1.1)

F (s, x) = LW {f (t, y )} =

∞ ∞

∫ ∫ f (t, y ) e

− st

( x − y )2

−

dy dt

The conventional Weierstrass transform
is defined by

4π 0 0

( 1.3)

2. The Testing Function Space LWa,b :

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ISSN 2229-5518

Let a and b be fixed numbers in 𝑅1 , let

⇒ f is R-integrable over any finite interval in

f (t, y ) variable in 𝑅1 , and let h

(t, y ) denote

the range (t, y ) ≥ 0 .

the function:

−𝒂𝒚

( x −

− st −

⇒ e

y )2

f (t, y )

𝒆 𝟐 , −∞ < 𝑦 < 0

4 is R-integrable over any

𝒉𝒂,𝒃 (𝒕, 𝒚) = �

−𝒃𝒚

𝒆 𝟐 , 0 ≤ 𝑦 < ∞
(2.1)

finite interval in the range (t, y ) ≥ 0 .

Now by definition of LW transform ,

LWa ,b as the linear space of all complex valued

smooth functions φ (t, y ) on 0 < t < ∞ ,

1 ∞ ∞

LW { f (t, y )} = ∫ ∫ f (t, y ) e

− st −

( x − y )2

4 dy dt

− ∞ < y < ∞ such that for each p, q = 0, 1, 2, -

4π 0 0

- - ∞ ∞

y2 ( x− y )2

−st −

at + y

≤ ∫ ∫ M ' eat e 4t0 e

0 0

4 dy dt

γ a ,b, p ,q

φ (t, y ) =

sup e

0<t <∞

0< y <∞

4 ha ,b

(t, y )D p + qφ (t, y ) < ∞

≤ − M ' e

− x 2

(2.2)

(s − a )x π

IJSER

The space LWa ,b is complete and a

Frechet space. This topology is generated by the

≤ M e

− x2

where- M ' = M

total families of countably multinorms space given by (2.2).

But

− x 2

Me 4

π (s − a )

is finite quantity.

3. Existence Theorem for LW Transform:

x π (s − a )

If f (t, y )

is a function which is
∴ LW {f (t, y )} exists provided, s > a and x > 0.
piecewise continuous on every finite interval in

the range (t, y ) ≥ 0

and satisfies

4. Theorem: If

LW { f (t, y )} = F (s, x) then

f (t, y )

y 2

≤ M ' eat e 4t0

. For all (t, y ) ≥ 0 and for

i) LW {t n f (t, y )}= (− 1)n

and

ds n

F (s, x)

some constant a, t0 and

M ' , then the LW

transform of

x > 0 .

f (t, y )

exists for all

s > a

and

ii) LW {(x − y )n f (t, y )}= (− 1)n (2)n

dx n

F (s, x)

Proof: It is given that the function

f (t, y ) is

Proof: Given that,

LW { f (t, y )} = F (s, x)

piecewise continuous on every finite interval in the range (t, y ) ≥ 0 .

i) By definition of LW transform

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 620

ISSN 2229-5518

F ( s, x ) = LW { f (t, y )}

{ ( )} ( )r +1

d r +1 ( )

∞ ∞

− st −

= ∫ ∫ f t, y e

0 0

( x − y )2

dy dt

(4.1)

LW t r +1 f t, y

= −1

dsr +1

F s, x

(4.4)

F ' ( s, x ) =

d {F ( s, x ) }

⇒ Theorem is true for n = r + 1.
Hence by induction , the theorem is true for all

d  1

∞ ∞

− st −

∫∫ f t, y e

=( x − y )2



dy dt

positive integral values of n.

 

ds 

4π 0 0



ii) Given that,

LW { f (t, y )} = F (s, x)

= − 1

4π

∞ ∞

∫ ∫ t f (t, y ) e

0 0

−st −

( x− y )2

dy dt

(4.2)

F (s, x) = LW { f (t, y )} =

Now,

1 ∞

4π ∫

∞

∫ f (t, y ) e

− st −

( x − y )2

dy dt

(− 1)F ' (s, x) = LW {t f (t, y )}

F ' (s,

x) =

d F (s,

x) =

d  1



∞ ∞

∫ ∫ f (t,

y ) e

− st − =( x − y )



dy dt 

LW { t f (t, y )} = ( −1) d

F ( s, x )

(4.3)

dx dx 

4π 0 0



(4.5)

ds = − 1 LW {(x − y ) f (t, y )}

IJSER

i.e. the theorem is true for n = 1.
Next, let this theorem be true for n = r, then we have

LW {( x − y ) f (t, y )} = ( −1) (2) d dx

F ( s, x )

(4.6)

LW {t r f (t, y )}= (− 1)r

ds r

[F (s, x)]

(− 1)r F r (s, x) = LW {t r f (t, y )}

i.e. the theorem is true for n = 1.

1 ∞

= 4π ∫

∞

∫ t r

f (t, y ) e

− st −

( x − y )2

dy dt

Next, let this theorem be true for n = r, then we
have
On differentiating both sides w. r. t. s we get,

LW {(x − y )r

f (t, y )}= (− 1)r (2)r

d F (s, x)

(−1)r F r +1 ( s, x )

 ∞ ∞

( x − y )2

 (− 1)r (2)r

d [F (s, x)] = LW {(x − y )r

f (t, y )}

= d  1

− st −

∫ ∫ e

4 t r f (t, y ) dy dt 

dx r

 

ds 

4π 0 0 

( −1)r +1 F r +1 ( s, x )

∞ ∞

( x − y )2

1 ∞

= 4π ∫

∞

∫ (x − y )r

f (t, y ) e

− st −

( x − y )2

dy dt

1 − st −

= ∫ ∫ t r +1 f t, y e 4

0 0

dy dt

4π 0 0

Now differentiating both sides w. r. t. x we get,

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 621

ISSN 2229-5518

( −1)r ( 2)r

r +1

 F ( s, x )

[5] Saitoh S : The Weierstrass Transform and an

d 

∞ ∞

∫ ∫ ( x − y )

− st −

f t, y e

=( x − y )2



dy dt

Isometry in the Heat Equation Applicable
Anal, (1983)

 

dx 

4π 0 0

 [6] Chelo Ferreira. Jose L. Lopez,

Approximation of the Poisson Transform for

= − 1 LW {(x − y )r +1 f (t, y )}

LW {(x − y )r +1 f (t, y )}= (− 1)r +1 (2)r +1 F r +1 (s, x)

(4.7)
large and small values of
the Transformation Parameter, Ramanujan J. (2012) DOI 10,1007/11139-012-9438-y
⇒ Theorem is true for n = r + 1.
Hence by induction ,the theorem is true for all positive integral values of n.

5. Conclusion:

In this paper we have introduced LW transform of a function f (t, y )and seen how it is exists. Also we have proved another theorem

which is useful to solve certain equation.

References :

[1] Zemanian A H : Generalized Integral Transformation, Interscience Publishers, New York (1968).
[2] Zemanian A H : Distribution Theory and Transform Analysis, McGraw-Hill, New York (1965).
[3] Pathak R S : A Representation Theorem for a
Class of Stieltjes Transformable Generalized
Function (1974).
[4] Srivastava H M : The Weierstrass-Laguerre
Transform, National Academy of Sciences, Vol 68 , No 3, pp 554-556,(1971)