International Journal of Research in Engineering and Applied Sciences (IJREAS) Available online at http://euroasiapub.org/journals.php Vol. 6 Issue 10, October - 2016, pp. 75~86 ISSN(O): 2249-3905, ISSN(P) : 2349-6525 | Impact Factor: 6.573 | Thomson Reuters ID: L-5236-2015

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, Department of 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.

The Laplace transform is invertible on a large class of functions. The inverse

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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 :

LWa,b 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, - - -

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by y2 at  2 4 p q  a,b, p,q t, y  sup e Dt Dy t, y  , 0t  0 y  (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). 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 1  Re x   2  . Let      x y2 1 st  s, x   t, y e 4 dy dt t, y D , where t, y and set      . Then for   4 0 0

' any fixed real numbers r and r’ with 0rr   & 0     r r' xh2 r r' xh2 1 sg 1 sg f g,h,e 4  s, xd d  f g,h, e 4  s, xd d     4 r r' 4 r r'

(3.1.1)

'' Where s    i  and xi ,&    are any fixed real numbers such that

' ' 1     2 and 1     2 . Proof:- The case t, y  0 is trivial. Let us consider t, y  0 for anyt, y . Since

is analytic for 휎 < 푅푒푠 < 휎 and ' ' and is an entire function, Fs, x 1 2 1  Re x   2  s, x then the r. h. s. side integral of the equation (3.1.1) exists. We first show that,

r r' xh2 1 sg  g,h    e 4  s, xd d, as a function of g,h which is belongs to 4 r r '

LW For this we consider, a,b .

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bh h2 ag  2 4 p q  a,b, p,q  g,h  Sup e Dg Dh  g,h 0g 0h

bh h2 r r ' xh2 ag  1 sg  Sup e 2 4 D p D q e 4  s, x d d g h     0g 4 r ' 0h r

p q Carrying the operator Dg Dh within the integral and summation sign, which is easily justified A due to smoothness of the integral, we get

bh h2 r r' xh2 ag  sg 2 4 1 p q 4  a,b, p, q g,h  Sup e Dg Dh e  s, xd d 0g   4 r r ' 0h =

r r' [xh2 h2 ] bh sag  1 p 4 2 Sup  s, x s e Pq x  hd d 0g   4 r r ' 0h (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

LWa,b . Therefore r. h. s. of (3.1.1) is meaningful.

Next, partition the path of integration on the straight line from s  ir to s  ir into m intervals,

' 2r ' ' ' ' 2r each of length and from x    ir to x    ir into n intervals each of length . m n

th ' ' th Let si    iri be any point in the i interval and x j    ir j be any point in the j interval. Consider,

2 m n x j h ' 1 si g  4r r   g,h   s , x  e 4   m,n  i j  mn  4 i1 j1  

2 m n x j h ' 1 si g  2r r    s , x  e 4    i j  mn   i1 j1  

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(3.1.3)

Operate f g,h to above equation (3.1.3) term by term, we get

2 x j h ' m n s g 1 i 4 2r r f g,h, m, n g,h  f g,h,  si , x j  e  i1 j1 mn

2 x j h ' m n s g 1 i 4 2r r   f g,h,e  si , x j  (3.1.4) i1 j1  mn

x h2 s g j 1 i 4 In view of the fact that f g,h,e  si , x j  is continuous function on 

 r    r and  r '   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 p and q ,  m, n g,h g,h converges uniformly to zero on 0  g   and 0 h   as m   and n   . Consider, by h2 ag  2 4 p q  a,b, p, q  m, n g,h g,h Sup e Dg Dh  m, n g,h g,h 0g oh

2 x h 2 2 p bh h m n k  j  ' p bh h ag  si g 2r r ag  1 2 4 p 4 1 2 4  Sup e  si , x j si  Pq x j  he  e (3.1.5) 0g  i1 j1 q0 mn 4 0h r r ' xh2 k sg s p P x  h e 4  s, x d d    q     r r ' q0

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

Notice that

bh h2 xh2 k ag   sg e 2 4 4  0 as g   and h   for Re s  a and Re x  b. q0

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Therefore, given 0, we can choose G and H so large that for all g  G and h  H ,

1 bh h2 xh2 r r ' k 1 ag  sg  k  e 2 4 4  s p P x  h  s, x d d      q      q0 4 3 r ' q0   r 

Which is finite quantity. Since (ty , ) 0 the right hand side is finite. 푎 Now for all , the magnitude of the second term on the right hand side of g  G and h  H   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

' 1  r r k  m n k 2r r ' s p P x  h  s, x d d s p P x  h  s , x     q        i  q  j   i j  3 ' mn r r q0  i1 j1 q0

(3.1.6)

m' & n' m  m' and n  n' Now choose 0 0 , so large that, for 0 0 , the above expression (3.1.6) is less

2  ' ' than . Thus, for all g  G and h  H and m  m0 and n  n0 , the r. h. s. of equation 3   g,h  g,h  (3.1.5) is less than  i.e. a,b, p,q  m,n     . Moreover, on 0  g  G, 0  h  H

' ' and  r    r,  r   r , the expression

bh h2 xh2 ag  k sg 1 2 4 p 4 e s Pq x  he  s, x 4 q0 is uniformly continuous function. Therefore, in view of equation (3.1.6), there exists an ' ' ' '   g,h  g,h  m1 and n1 such that for all m  m1 and n  n1 , a,b, p, q  m, n     on

0  g  G and 0  h  H as well. Thus when m  max m' ,m' and n  max n, ,n, ,   0 1   0 1 

0  g    a,b, p,q m,n g,h g,h on  and 0  h  .  This completes the proof.

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3.2 Lemma 2:

  x y2 1 st  s, x  t, ye 4 dy dt For t, y D , set  as in lemma (1) then, 4 0 0

'  h  y  2 2 2 sin r ' xh y h   1 r r sg 1    sin rt  g ' h  y 2 e 4  s, xd d  e 4 4 t, ye tg  e   dy dt 4 r r '  0 0 t  g 2  h  y     2 

Proof: We shall prove the result by justifying the steps in the following manipulations and by considering compact support of  t, y D 푎 xh2 xh2   x y2 1 r r ' sg 1 r r' sg  1 st  e 4  s, xd d  e 4  t, ye 4 dy dt  d d r r' r r'  4 4  4 0 0 

'  h  y  2 2 2 sin r ' xh y h   1 r r sg 1    sin rt  g ' h  y 2 e 4  s, xd d  e 4 4 t, ye tg  e   dy dt 4 r r '  0 0 t  g 2  h  y     2 

3.3 Lemma 3:

' ' ' Let a1,a2 ,b1,b2 ,, ,r and r be real numbers with a1    a2 and b1    b2 . Also let    t, y  D   .

'  h  y  y2 h2 sin r   1    sin rt  g ' h  y 2 e 4 4 t, ye tg  e   dy dt  Cg,h Then,  0 0 t  g 2  h  y     2 

Converges in LW to g,h as r,r '   . a,b   

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Proof: Here we have to prove that

'  h  y  y2 h2 sin r   1    sin rt  g ' h  y 2 e 4 4 t, ye tg  e   dy dt  g,h  0 0 t  g 2  h  y     2  For this we show that

 a,b, p, q Cg,hg,h  0

Now consider,

 a,b, p, q Cg,hg,h

 '  h  y   bh h2    y2 h2 sin r    ag  1  sin rt  g ' h  y 2  Sup e 2 4 D p D q  e 4 4  t, y e tg  e   dy dt  g,h  0 g h      0g  0 0 t  g 2  h  y   0h       2  

 h  y  sin r '   1   sin rt  g ' h  y 2 t, ye tg  e   dy dt  g,h  0 0 t  g 2  h  y     2  (3.3.1)

(3.3.2)

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

 a,b, p, q Cg,hg,h

 '  h  y     y 2 h2 sin r      1  tg  sin rt  g  ' h  y  2    e 4 4 t, ye e dy dt  0 0 t  g 2  h  y   bh h2     ag   2  2 4 p q    Sup e Dg Dh 0g  '  h  y   0h    '  h y  sin r    1 sin rt  g    2  t, ye tg  e  2    dy dt     t  g  h  y   0 0    2     

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y 2 h2 '   sin rt ' sin r y  4 4 t1 1  y1 1 bh h2    e t1  g, h  2y1 e e dy1 dt1  ag   2 t y  Sup e 2 4 D p D q  1 1  g h      '  0g   2 sin rt ' sin r y 0h    t  g, h  2y et1 1 e y1 1 dy dt     1 1  1 1    t1 y1 

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

 Cg,hg,h  0 a,b, p, q . This completes the proof.

IV Theorem: 4.1 Inversion Theorem:

' Let Fs, x  LWf t, y for s, x  . Also let r and r are real variables. Then in the  f   sense of convergence in D’

 ir  ' ir' x y2 1 st f t, y  lim Fs, xe 4 dx ds (4.4.1) r, r'    4   ir  ' ir'

' Where  and  are any fixed positive real nos. belonging to  f , 1     2 and

' ' . 1     2 .

Proof: We need to show that for t, y D.

 ir  ' ir' x y2 1 st 1   Fs, xe 4 dx ds, t, y  f g,h, g,h (4.4.2) 4   ir  ' ir' 2 ' as r,r  

From the analyticity of Fs, x on  f and the fact that the t, y has compct support

퐴 퐴 퐴 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

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

   ir  ' ir' x y2  1 st  t, y   Fs, xe 4 dx ds dy dt 0 0 4   ir  ' ir' 

Letting s    i and x   '  i we get,

  r r ' x y2 1 st t, y  Fs, xe 4 d d dy dt 4  0 0 r r '

Since Φ(t,y) has a compact support and integrand is a continuous function of t, y,,, the

퐴 order of integration may be changed. These yields

r r '   x y2 st   Fs, x t, ye 4 dy dt d d ' r r 0 0

r r ' xh2   x y2 1 sg st    f g,h,e 4 t, ye 4 dy dt d d r r ' 4 0 0

 '  h  y    y2 h2 sin r    1 1    sin rt  g ' h  y 2  f g,h,  e 4 4 t, ye tg  e   dy dt 2  0 0 t  g 2  h  y      2     

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

1 f g,h, g,h 2 1 = f g,h, g,h 2 Which complete the proof of the inversion theorem.

4.2 Uniqueness Theorem:

Let LWf t, y Fs, x for s, x f and LWut, y Us, x for s, xu and if

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 f  u is not empty. If Fs, x  Us, xon , 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

 ir  ' ir' x y2 1 st f t, y ut, y, t, y  lim Fs, xUs, xe 4 dx ds,t, y r, r '    4   ir  ' ir'

(4.2.1)

But given that

Therefore equation (4.2.1) becomes

 0, t, y D

Hence in the sense of equality in .

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] Mathurkar S. 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. [7] Thakur A. K. & Tamrakar P., 2015, Convergence and inversion theorems for Generalized Weierstrass transform, Bulletin of Mathematical Sciences and

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Applications, ISSN: 2278-9634, Vol.13, pp. 6-12. [8] Zemanian A. H., 1968 : Generalized Integral Transformation, Interscience Publishers, New York.

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