On Generalized of Laplace-Weierstrass Transform

ISSN(Online) : 2319-8753 ISSN (Print) : 2347-6710

International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization)

Vol. 5, Issue 10, October 2016

V. N. Mahalle1, S.S. Mathurkar2, R. D. Taywade3 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, Department of 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 x y 2 st defining testing function space LW Laplace-Weierstrass transform is defined by taking e 4 as a kernel, as it a,b . 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 transform is 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 x y 2 st of f t, y to e 4 x y 2 1 st  i.e. Fs, x  f t, y, e 4 4

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International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization)

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x y 2 st 4 For this purpose we construct a testing function space LWa, b which contains the kernel e for all f t, y in some restricted domain.

II. TESTING FUNCTION SPACE

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, - - - by y 2 at  2 4 p q  a,b, p,q t, y  sup e Dt Dy t, y  , (2.1) 0t  0 y  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

2 If, on the half plane {s :a  Re s}and { x :b  Re x }, Gs, x is analytic and satisfies Gs, x  K1 K 2 s, x ,

 i  ' i x y 2 1 st  where K , K are constants and if g t, y  G s, x e 4 dx ds , a   & b   ' 1 2       4   i  ' i (3.1) then is a continuous function that does not depend on the choice of & ' and generates a regular gt, y   * generalized function in LWa,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) 2 2  '  y    i '  y t 1 it e 4 gt, y    G  i, '  i ' e 4 d d ' , a  & b   ' 4   (3.2) In view of the bound on Gs, x it follows that the last integral converges uniformly for allt, 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 LWa,b . Finally from inversion theorem it also follows that Gs, x  LWgt, y for a  Re s and b  Re x . This completes the proof of the lemma.

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IV. CHARACTERISTIC THEOREM FOR LAPLACE-WEIERSTRASS TRANSFORM

Theorem: If for, ' ' , is bounded Fs, x  LWf t, y s, x  f  s, x/ 1  Re s   2 ,  1  Re x   2  Fs, x 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 ' .

x y 2 1 st  Fs, x  f t, y,e 4 4

2 ' x y   st  K 4  max  a,b, p,q e  0 pr 4 '   0qr   2 by y 2 x y   at  st  2 4 p q y  K max Sup e Dt Dy e  0 pr ' 0t   0qr 0 y   by y 2 x y 2   p sat 2 4 4  K max s pq x  y Sup e .e 0 pr ' ot 0qr 0 y Fs, x  K Ps x  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 Fs, xe x K K  1 2 , a  Re s and b  Re x Qs, x s, x 2 where K1, K 2 are constants. 2 x2  i  ' i x y  Fs, xe 1 st  Set Gs, x  . Then, gt, y    Gs, xe 4 dx ds Qs, x 4  '  i  i , ' a   and b  

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Satisfies conditions of above lemma and is a member of LW * . Now let f t, y  QDgt, y where D a,b , represent the generalized differentiation in LW * . Then f t, y LW * a,b is also a member of a,b and LWf t, y Qs, xGs, x  Fs, x for a  Res and b  Re x .

V. REPRESENTATION THEOREM FOR LAPLACE-WEIERSTRASS TRANSFORM

Theorem: * Let f t, y be an arbitrary element of LWa,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 hm, n t, y defined over  such that,

by y2 r1  1 at  mn 2 4 m n f ,  1 e Dt Dy hm,n t, y,t, y , m0 n0

where r and  are appropriate non-negative integers satisfying m  r 1 and n  1   Proof: Let a,b, p,q p, q0 be the sequence of seminorms. Let f t, y be an arbitrary element of LW * and t, ybe an element of D .Then by the a,b boundedness property of generalized functions we have for an appropriate constant K and a non negative integers r and  satisfying p  r and q 

f ,  K max  a,b, p,q t, y p r q  by y 2 at  2 4 p q  K max Sup e Dt Dy t, y p r 0t q  0 y by y 2 at  ' 2 4 m n  K max Sup e max Dt Dy t, y p r 0t m p nq q  0 y where K’ is a constant which depends only on m, n and hence p, q by y 2 at  '' 2 4 m n f ,  K max Sup e Dt Dy t, y (5.1) mr 0t n 0 y Now let us set, by y2 at   t, y  e 2 4 t, y, m  r, n  (5.2) r, Then clearly,  t, y D r, Also,

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by y 2 at  t, y  e 2 4  t, y r, On differentiating above equation with respect to t and y successively we get, by y 2    at   D D t, y   e 2 4  t, y t y r, y t   2   r, t,y  b  y  r, t, y  by y2      at   y t  2  t   e 2 4      r, t, y  b  y    a  a t, y   y r, 2     Let us suppose that in  , Sup  Sup    t, y/ C '  t  D' , A'  y  B '  r, 2 Then since, by y 2 at  2 4 e  0 by y 2 2 at    t, y  t, y   t, y  2 4  b  y b  y r, r, r,  Dt Dy  e   a r, t, y    a    2 2 t y y t 

by y 2 ' ' 2 at    t, y  t, y   t, y  2 4  b  A b  A r, r, r,   e  a r, t, y   a    2 2 t y y t  by y2 2 at    t, y  t, y   t, y  ''' 2 4  r, r, r,   K e r, t, y      t y y t  ' ' '''  b  A b  A  where K  max  a , , a ,1  2 2  iv Hence by induction we can prove that in  for obvious constant K , which depends on a and b then, by y2 at  D m D nt, y  K iv e 2 4 D c D d  t, y (5.3) t y  t y r, cm dn Using equation (5.3) in equation (5.1), we get by y 2 by y2 at  at  '' 2 4 iv 2 4 c d f ,  K max Sup e K e  Dt Dy r, t, y mr 0t n cm 0 y d n v c d  K max Sup Dt Dy r, t, y (5.4) mr 0t n 0 y where c  m and d  n are suitable constants.

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Now we can write,   Sup t, y  Sup D D t, ydt dy  t y 0t 0t t y 0 y 0 y  D D t, y (5.5) t y L'L' Hence from equation (5.1), equation (5.4) becomes, vi m n f ,  K max Sup Dt Dy r, t, y ' ' (5.6) mr1 0t LL n 1 0 y 2 Let the product space ' ' be denoted by ' L  L L  . We consider the linear one to one mapping m n ' 2  :  Dt Dy t, ymr1 D L  n 1 of into . In view of equation (5.6) we see that the linear function 2 :  f , D L' r, is continuous on for the topology induced by   . Hence by Hahn Banach theorem, it 2 2 L' L' L can be a continuous linear functional in the whole of   . But the dual of   is isomorphic with , therefore L h m  r 1, n  1 there exists - function m, n such that, f ,  h t, y, D m D n  t, y  m,n t y r, mr1 n 1 From equation (5.2) we have, by y 2 at  f ,  h t, y, D m D n e 2 4 t, y  m,n t y mr1 n 1 Using property of differentiation of a distribution by an infinitely smooth function, we get by y 2 at  f ,  1mn D m D n h t, ye 2 4 , t, y  t y m,n mr1 n 1 Now by using property of multiplication of a distribution by an infinitely smooth function, we get by y 2 at  f ,  1mn e 2 4 D m D n h t, y, t, y  t y m,n mr1 n 1

V. CONCLUSION

In the present paper the boundedness theorem and representation theorem for the Laplace-Weierstrass are proved

ISSN(Online) : 2319-8753 ISSN (Print) : 2347-6710

International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization)

Vol. 5, Issue 10, October 2016

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] Mathurkar S. 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. [9] Zemanian A. H., “ Generalized integral transformations”, Pore and Applied mathematics, Vol.XVIII, Interscience Publishers, New York London- Sydney, 1968.