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75 Inversion Theorem for Laplace-Weierstrass Transform 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 International Journal of Research in Engineering & Applied Sciences Email:- [email protected], http://www.euroasiapub.org 75 An open access scholarly, online, peer-reviewed, interdisciplinary, monthly, and fully refereed journals International Journal of Research in Engineering and Applied Sciences (IJREAS) Vol. 6 Issue 10, October - 2016 ISSN(O): 2249-3905, ISSN(P) : 2349-6525 | Impact Factor: 6.573 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, - - - International Journal of Research in Engineering & Applied Sciences 76 Email:- [email protected], http://www.euroasiapub.org An open access scholarly, online, peer-reviewed, interdisciplinary, monthly, and fully refereed journals International Journal of Research in Engineering and Applied Sciences (IJREAS) Vol. 6 Issue 10, October - 2016 ISSN(O): 2249-3905, ISSN(P) : 2349-6525 | Impact Factor: 6.573 by y2 at 2 4 p q a,b, p,q t, y sup e Dt Dy t, y , 0t 0 y (2.1) for some fixed numbers a,b in R The space 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 y2 1 st s, x t, ye 4 dy dt t, y D , where t, y and set . Then for 4 0 0 ' any fixed real numbers r and r’ with 0rr & 0 r r' xh2 r r' xh2 1 sg 1 sg f g,h,e 4 s, xd d f g,h, e 4 s, xd 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 anyt, y . Since is analytic for 휎 < 푅푒푠 < 휎 and ' ' and is an entire function, Fs, 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' xh2 LW 1 sg ga,,hb e 4 s, xd d, as a function of g,h which is belongs to 4 r r ' LW For this we consider, a,b . International Journal of Research in Engineering & Applied Sciences 77 Email:- [email protected], http://www.euroasiapub.org An open access scholarly, online, peer-reviewed, interdisciplinary, monthly, and fully refereed journals International Journal of Research in Engineering and Applied Sciences (IJREAS) Vol. 6 Issue 10, October - 2016 ISSN(O): 2249-3905, ISSN(P) : 2349-6525 | Impact Factor: 6.573 bh h2 ag 2 4 p q a,b, p,q g,h Sup e Dg Dh g,h 0g 0h bh h2 r r ' xh2 ag sg 2 4 p q 1 4 Sup e Dg Dh e s, xd d 0g 4 r r ' 0h 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' xh2 ag 1 sg g,h Sup e 2 4 D p D q e 4 s, x d d a,b, p, q g h 0g 4 r ' 0h r = r r' [xh2 h2 ] bh 1 sag Sup s, x s p e 4 2 P x h d d q 0g 4 r ' 0h r (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,hbelongs 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 4 m,n g,h si , x j e 4 i1 j1 mn 2 m n x j h ' 1 si g 2r r 4 si , x j e i1 j1 mn International Journal of Research in Engineering & Applied Sciences 78 Email:- [email protected], http://www.euroasiapub.org An open access scholarly, online, peer-reviewed, interdisciplinary, monthly, and fully refereed journals International Journal of Research in Engineering and Applied Sciences (IJREAS) Vol. 6 Issue 10, October - 2016 ISSN(O): 2249-3905, ISSN(P) : 2349-6525 | Impact Factor: 6.573 (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 i1 j1 mn 2 x j h ' m n s g 1 i 4 2r r f g,h,e si , x j (3.1.4) i1 j1 mn x h2 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 0g oh 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 he e (3.1.5) 0g i1 j1 q0 mn 4 0h r r ' xh2 k sg s p P x h e 4 s, x d d q r r ' q0 Where Pq is the polynomial in q and q is finite quantity. Notice that bh h2 xh2 k ag sg e 2 4 4 0 as g and h for Re s a and Re x b.
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