P: ISSN No. 0976-8602 RNI No.UPENG/2012/42622 VOL.-V, ISSUE-III, July-2016 E: ISSN No. 2349 - 9443 Asian Resonance The Weierstrass Transform on Some Spaces of Type S Abstract The main objective of this paper is to study the classical theory of the Weierstrass transform is extended to the Dirac delta function and some of their basic properties. The characterizations of the Weierstrass transform on the Gel'fand-Shilov spaces of type S is shown. Keywords: Weierstrass transform, Fourier transform, Green's function, Dirac delta function. Introduction This present paper, my contribution is to define the Weierstrass transform in the form of Dirac delta function and study some of their basic properties. The characterizations of the Weierstrass transform on the Gel'fand-Shilov spaces of type S is shown. The conventional Weierstrass transform of a suitably restricted function fx() on the real axis R is defined by [5, p. 267] as: 1 (tx )2 (1) W f( x ) f ( x )exp dx . 2 R 4 The Green's function G t x, k of the heat equation for the infinite interval is 1 (tx )2 , where k is positive constant. G t x, k exp 2 k 4k Ashutosh Mahato The limit of as 0 represent the Dirac delta Assistant Professor, 2 function 1 (tx ) Deptt.of Mathematics, (tx ) lim exp . C.V.Raman College of 0 2 k 4k Engineering, Bhubaneswar, So, in terms of Dirac delta function ()tx , the Weierstrass Odisha transform (1) can be rewritten as Wfx ()(,)()(). fxtˆ fx txdx (2) R Aim of the Study/ The problem / Objective of the Study The main objective of this paper is to develop the classical theory of the Weierstrass transform is extended to the Dirac delta function and some of their basic properties. The characterizations of the Weierstrass transform on the Gel'fand-Shilov spaces of type S, that is, S , S and S is shown. The Weierstrass transform arises naturally in problems involving the heat equation for one dimensional flow. The Weierstrass transform of certain class of generalized functions which are duals of so-called testing function spaces and . Wab, W , Review of Literature The Weierstrass transform arises naturally in problems involving the heat equation for one dimensional flow. The Weierstrass transform of certain class of generalized functions which are duals of so-called testing function spaces and introduced by Zemanian [6]. The inversion formulas were also obtained. On the other hand the Weierstrass transform of bounded functions and certain other functions with prescribed growth conditions are all characterized by Hirshman and Widder [3]. Karunakaran and Vnugopal [4] introduced the Weierstrass transform on the generalized function , obtained an inversion formula, and also characterize the Weierstrass transform of elements from both the testing function spaces and ' . In this present paper, motivated by the work of 108 P: ISSN No. 0976-8602 RNI No.UPENG/2012/42622 VOL.-V, ISSUE-III, July-2016 E: ISSN No. 2349-9443 Asian Resonance Zemanian [6], and start with the Gel'fand-Shilov Wfxa[( )] fxa ( )(( t xadx )). spaces of type S, that is, , and defined by R Gel'fand and Shilov [2] and develop some Substituting xa, then the right-hand side of characterization of Weierstrass transform on these the above expression becomes spaces. Properties of the Weierstrass Transform f()() t d Relation between the Fourier transform and the R Weierstrass Transform 1 i(-) t u f() e du d The Fourier transform of fx()is defined by RR2 F f()()() x y f y itu1 - it e e(-)() t f d dt du (3) 1 RRR2 eiyx f(), x dx e-it W[ f ( )] ( t ) dt 2 R where the kernel eiyx of the Fourier transform can be R -itp -iyx it pW[ f ( )] e ( tp ) dt e e () t y dt R written as . R pfˆ( t - a ). So that,equn.(3) becomes The Spaces of Type S fy() The spaces of type S, that is, , and Play an important role in the theory of linear 1 it . e ()() t y f x dxdt partial differential equations as an intermediate 2 RR spaces between the spaces of the C and the analytic Substituting y sx and t sp where s is functions. The Fourier transform has been studied on the spaces of type S by Friedman [1] and Gel'fand real, in the above expression, we and Shilov [2]. The aim of this paper is to study the obtain characterizations of the Weierstrass transform (2) on 1 isp these spaces. Let us recall the definitions of these f()()() sx e s f x sp sx dx dp 2 spaces. RRDefinition 1 1 isp The space consists of all e fxpdpˆˆ(,)(,)(). Ffxp FWf S , 0, 2 R infinitely differentiable functions(xx ), , This means that f is the Fourier transform satisfying the inequalities q d of f , whereas F[ W ( f )]is the Fourier transform of ()() Sup xk x kq, the Weierstrass transform of . Thus xR dx kk fˆ( x , p ) W [ f (.)]. Cq A k, k , q N0 , (4) where the constants A and dependS on Sthe functionS Cq Theorem1 (Linearity) k 1 . For k=0, the expression k is considered to be If f( x ), g ( x ) L (R) and a, b are any scalars, equal to 1. then Wafx[() bgx ()] aWfx [()] bWgx [()] Definition2 Proof The space S , 0,consists of all infinitely By definition, we have differentiable functions , satisfying Wafx[() bgx ()] [() afx bgx ()]( t xdx ) the inequalities R q afx()()()() t xdxbgx t xdx k d kq, ()() Sup x x RR x dx aW[ f ( x )] bW [ g ( x )]. C Bqq q , k , q , (5) Theorem2 (Shifting) k 0 where the constants Band depend on the function IfW[ f ( x )] fˆ ( t , x ), then Ck W[ f ( x a )] pfˆ ( t a ). Proof By definition, we have 109 P: ISSN No. 0976-8602 RNI No.UPENG/2012/42622 VOL.-V, ISSUE-III, July-2016 E: ISSN No. 2349 - 9443 Asian Resonance Definition3 q q kk kkd 1 q k k x W f( x ) m !2 The space S , , 0, consists of all dx 2 k0k m 0 m u 0 u infinitely differentiable functions , k d dx dy Sup xu f(). x satisfying the inequalities qq x dx(1 | x |) (1 | y |) q k d Now, using the Definition of (4) and the property (8), kq, ()() Sup x x x dx we can derive the above expression as CAk B q k k q q , k , q , (6) 1 q q kk k k 0 q!2q C A u u u where the constants A, Band depend on the k C 2 k0k m 0 m u 0 u function . dx dy The spaces of type S are closely interrelated 4. qq by means of the Fourier transform; namely, the 00(1 |xy |) (1 | |) formulae The integral can be made convergent by ˆ ˆ ˆ S S, S S and S S hold. (7) choosing q 1, then the right-hand side of the above We shall make use of the following expression can be bounded by q kk inequalities in our investigation: 2 q q q k k u q!2 q Ck A q k m u q! qq q k0 m 0 u 0 k! k ! k !2 , (8) q k 2 q q qk k (qk )! kkk0 q!2 q Ck (1 A ) ()k q k q k q km00km and (9) (),,k q k q e e k q 0 q 2 q q qq k Theorem 3 q!2 q (1 A ) Ck 2 The Weierstrass transform in terms of Dirac k 0 k delta function is defined by (2) is a well-defined '3q q Ck q2 (1 A ) mapping of into S for 0. ' qq Ck B q . Proof This completes the proof of the theorem. Let kq, 0 , we obtain Theorem 4 qq The Weierstrass transform in terms of Dirac dd delta function is defined by (2) is a well-defined xkk W f()()() x x f x t x dx (xx ), dx dx mapping of S into S for 0. q Theorem 5 kd 1 i() t x y x f() x e dy dx The Weierstrass transform in terms of Dirac dx 2 delta function is defined by (2) is a well-defined q mapping of S into S for , 0. 1 kd i() t x y x f() x e dxdy 2 dx The proof of the above theoremsS are similar as the proofof Theorem 3. q k k References 1 q ddq2 k ixy ( q k ) ity f()() x i e y e dy dx. 1. Friedman, A (1963), Generalized Functions and 2 k 0 k dx dy Partial Differential Equations, Prentice-Hall, Englewood Cliffs, NJ Using integration by parts, the right-hand 2. Gel'fand, I. M and Shilov, G. E (1968), Vol.2, side of the last expression can be estimated by Generalized Functions, Academic Press, New York q k k and London. 1!qk dk3q 4 k ixy ( k m ) ity f()()(). x i e ity e dy dx 3. Hirshman, I. I and Widder, D. V (1955), The 2 km00km dx ( k m )! Convolution Transform, Princeton Univ. Press, Therefore, Princeton, NJ. 4. Karunakaran, V and Venugopal, T (1998), The Weierstrass Transform for a Class of Generalized Functions, Journal of Mathematical Analysis and Applications, Vol. 220, pp 508-527. 5. Lokenath Debnath, (2002), Wavelet Transforms and Their Applications, Birkhäuser, Boston. 6. Zemanian, A. H (1968), Generalized integral transformations, Wiley-Interscience Publishers, NewYork 110 .