IOSR Journal of Mathematics (IOSRJM) ISSN: 2278-5728 Volume 1, Issue 4 (July-Aug 212), PP 19-23 Aelian Theorem for Generalized Mellin-Whittaker Transform A. S. Gudadhe 1, R. V. Kene 2 1, 2 (Head Department of Mathematics, GVISH/ Amravati, M S, India) Astract: The paper provides the Aelian and Tererian theorem for generalized Mellin-Whittaker transform. Key-words:Aeliantheorem, Generalized Mellin-Whittaker transform. I. Introduction: Integral transformation is one of the well known techniques used for function transformation. Integral transform methods have proved to e of great importance in initial and oundary value prolems of partial differential equation. Extension of some transformations to generalized functions has een done time to time and their properties have een studied y various mathematicians. However there is much scope in extending doule and triple transformation to a certain class of generalized functions. Bhosale B. N. and Choudhary M. S. [1], Sharma V. D. and Gudadhe A. S. [2] has discussed doule transforms. Motivated y this we have also defined a new comination of integral transforms on the spaces of generalized functions namely Mellin-Whittaker transform. Along with the definition its analyticity theorem is proved in [3], where as inversion theorem is investigated in [4]. Now in the present paper we estalished the aelian theorem for generalized Mellin-Whittaker transform. In section 2 we have defined the spaces E α, and E,w on which Mellin transform is defined. We have given the definition of generalized Mellin Whittaker transform in section 3. Section 4 is devoted toinitial value theorem for generalized Mellin-Whittaker transform. Finally in section 5 final value theorem for generalized Mellin-Whittaker transform is estalished. Notations and terminology as per Zemanian [5]. II. Spaces E α, and E,w : By E,c (, c are finite real numers with < c) we denote the linear space of infinitely differentiale functions φ(x) defined on [, ] and such that there exist two strictly positive numers ζ and ζ for which x k+1 ζ φ k x as x + and x k+1 c ζ φ k x as x for all k N, where N is the set of non negative integers. We set x s 1 + = xs 1, x >, x < so that x + s 1 elongs to E,c if < Res < c. Put K,c x = x, < x 1 x c x 1 and γ k,,c φ = sup x> x,cx k+1 φ k (x). γ k,,c φ are all ounded and are seminorms, γ,,c is a norm. We now provide the following topology in E,c. A sequence φ j in E,c if and only if γ k,,c (φ j ) for each k N.ThusE,c is provided with a structure of a countale multi-normed space. Also, E,c is the inductive it of E,c as. This means that a sequence φ j in E,c if and only if there exist a < c such that φ j E,c and φ j in E,c (i.e. γ k,,c (φ j ) as j ). In a similar manner, E α, is the inductive it of E,c as α, c and E,w is the inductive it of E,c as, c w. E α, is the dual of E α, and E,w is the dual of E,w. III. Generalized Mellin Whittaker Transform: For f(x, t) MW a,, we define distriutional Mellin Whittaker transform of a function f(x, t) as, MW f x, t = F s, y = f x, t, x s 1 e q 2 yt yt m 1 2W k,m (pyt) (3.1) RHS of (3.1) has meaning, for f MW a, and x s 1 e q 2 yt yt m 1 2W k,m (pyt) MW a,. IV. Initial Value Theorem For Generalized Mellin-Whittaker Transform: In this section we have proved initial value theorem for generalized Mellin-Whittaker transform., Theorem: Let f(x, t) e locally integrale function with distriutional derivatives w.r.to x elonging to E α, for some real numer α and having compact support for x in, a. Moreover f(x,t) e ct f(x,t) x a t log a v x t η is asolutely integrale for c R. = β Rev > 1, η > 1. Then 19 Page
s v+1 y η +1 F(s,y ) s y a s v+1 B m,η,m,k,q,p whereb m, η, m, k, q, p = = βin the half plane Res > α, p m +1 2Γ m ±m +η +1 1 2 q+p m +m +η +1 Γ(m k+η + 3 2 ) 2F 1 Aelian Theorem for Generalized Mellin-Whittaker Transform m + m + η + 1, m k + 1 2 ; q p q + p m k + η + 3 2 ; Proof: We prove the theorem for a restricted type of functions i.e. when f(x, t) is seperale. Let f x, t = g x (t) (4.1) Therefore we can assume that β = β 1. β 2, where β 1, β 2 are real numers such that, property (ii) ecomes x a g x log a x v = β 1 Re v > 1, (4.2) t t = β t η 2 η > 1 If we put x = ae V then (4.2) and (4.3) x a t orv g ae V (t) t V g x (t) log a x = β 1. β 2 = β g(ae V ) V v = β 1 (4.3) v t η = β 1. β 2 = β We know that M g x s = a s Lg ae V (s). (4.4) Also for η > 1 and t > t η e q 2 yt yt m 1 2W k,m pyt dt = 1 η +1 B(m, η, m, k, q, p) V v e s dv Now consider, = (v+1) sv+1 forv > 1 (4.6) a y (4.5) s v+1 y η +1 F s, y β (v + 1) a s B(m, η, m, k, q, p) = s v+1 y η +1 x s 1 e q 2 yt yt m 1 2W k,m pyt g x t dxdt β 1. β 2 v + 1 a s B(m, η, m, k, q, p) (4.7) f x, t has compact support in [, a] w.r.t. x. Put x = ae V x s 1 dx = a s e sv dv Also f x, t = g x t = g ae V (t) and as x varies from to a and V varies from to. Therefore asoring negative sign and using (4.5), (4.6) we have LHS of (4.7) as, LHS = s v+1 η +1 a β 1. β 2 s v+1 q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t dvdt V v e sv dv a s η +1 y t η e q 2 yt yt m 1 2W k,m pyt dt = s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t β 1. β 2 dvdt = s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t β 1. β 2 dvdt I 1 + I 2 (4.8) where As W k,m z = O z k e z 2, q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t β 1. β 2 dxdt q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t β 1. β 2 dxdt z = O (z m +1 2), z, m < 2 Page
Aelian Theorem for Generalized Mellin-Whittaker Transform = O (z m 1 2), z, m > The function s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt a s e sv is ounded as s and y. That is s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt a s e sv M as s and y for < x < and < t <. Therefore I 1 ε sup <x< <t< M 2 g ae V t β 1. β 2 for ε = 2M2 > there exists δ > such that x < δ 1, t < δ 2 g ae V t β 1. β 2 < ε 2M 2 I 1 < ast, V using (3), for all >. (4.9) 2 Now for q 2 yt yt m 1 2W k,m pyt a s e sv g ae V t β 1. β 2 dvdt By the property (i) f(x,t) is asolutely integrale hence g ae V t is asolutely integrale and I e ct e ct 2 can e written as, 1 2 (qy 2c)t yt m 1 2W k,m pyt a s e sv e ct g ae V t β 1. β 2 dvdt Now the function e sv e 1 2 (qy 2c)t yt m 1 2W k,m pyt is finite and continuous in < T < and it as t if Re q + p x > 2c. Let the upper ound of this function e attained at the point V, t = (r 1, r 2 ) and g ae V t β 1. β 2 dvdt = M 1 say then I 2 M 1 e sr 1e 1 2 (qy 2c)r 1 yr 2 m 1 2W k,m pyr 2 s v+1 y η +1 (4.1) Choose y sufficiently small so that r.h.s. of (4.1) ecomes less than ε 2. Hence (4.8) ecomes Hence the theorem is proved. s v+1 y η +1 F s, y β (v + 1) a s B(m, η, m, k, q, p) I 1 + I 2 < ε 2 + ε 2 = ε V. Final Value Theorem For Generalized Mellin-Whittaker Transform: In this section we estalished final value theorem for generalized Mellin-Whittaker transform. Theorem: Let f(x, t) e locally integrale function with all distriutional derivatives w.r.to x elonging to,, for some real numer w and having compact support for x in [, ). Moreover E,w f(x,t) f(x,t) x t log x v t η s v+1 y η +1 F(s,y ) s y s Γ v+1 B m,η,m,k,q,p is asolutely integrale for some c R. = β Rev > 1, Reη > 1. Then whereb m, η, m, k, q, p = = βin the half plane Res > α, p m +1 2Γ m ±m +η +1 1 2 q+p m +m +η +1 Γ(m k+η + 3 2 ) 2F 1 m + m + η + 1, m k + 1 2 ; q p q + p m k + η + 3 2 ; Proof: We prove the theorem for a restricted type of functions i.e. when f(x, t) is seperale. Let f x, t = g x (t) (5.1) Therefore we can assume that β = β 1. β 2, where β 1, β 2 are real numers such that, Property (ii) ecomes x g x log x v = β 1 Re v > 1,(5.2) t t = β t η 2 η > 1 We also know that M g x s = g x x s 1 dx Since g(x) has compact support in [, ) we have, 21 Page
Aelian Theorem for Generalized Mellin-Whittaker Transform = g x x s 1 dx If we put x = e V in aove equation = g e V s e sv dv = s g e V e eiπ sv dv = s Lg e V (e iπ s). (5.3) Also for η > 1 and t > t η e q 2 yt yt m 1 2W k,m pyt dt = 1 η +1 B(m, η, m, k, q, p) (5.4) V v e sv dv y = Γ(v+1) ( 1) v+1 sv+1forrev > 1 (5.5) Now consider, s v+1 y η +1 F s, y β s Γ(v + 1)B(m, η, m, k, q, p) = s v+1 y η +1 x s 1 e q 2 yt yt m 1 2W k,m pyt g x t dxdt β 1 β 2 s Γ(v + 1) B m, η, m, k, q, p ( g(x)has compact support in,. )(5.6) Put x = e V x s 1 dx = s 1 e V(s 1). e V dv = s e V dv As x varies from to and V varies from to. Using (5.4), (5.5) and the suggested sustitution (5.6) ecomes s v+1 y η +1 F s, y β s Γ(v + 1)B(m, η, m, k, q, p) = s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt s e sv g e V t dvdt β 1. β 2 ( 1) v+1 s v+1 s V v e sv dv y η +1 t η e q 2 yt yt m 1 2W k,m pyt dt = s v+1 y η +1 e I 1 + I 2 (5.7) where Consider Using the asymptotic ehaviour of W k,m z = O z k e z 2, As f(x,t) = g x (t) Therefore Therefore q 2 yt yt m 1 2W k,m pyt s e sv g ev t z = O (z m +1 2), z, m < = O (z m 1 2), z, m > = g ev (t) c e Vc is asolutely integrale for some c R. g e V (t) e Vc is asolutely integrale as c is constant. q 2 yt yt m 1 2W k,m pyt s e (s+c)v g ev t Therefore e cv g ev t β 1. β 2 dvdt = M Let (r 1, r 2 ) e the point where the r.h.s. function has upper ound M. sup < V < s v+1 y η +1 e q 2 yr 2 yr m 1 2 2W k,m pyr 2 s e s+c r 1 M. < t < Choose y sufficiently small that r.h.s. < ε. (5.8) 2 Consider β 1. β 2 dvdt β 1. β 2 dvdt 22 Page
Aelian Theorem for Generalized Mellin-Whittaker Transform s v+1 y η +1 e q 2 yt yt m 1 2W k,m pyt s e sv is finite and continuous in < t < and < x < and it converges to zero as t and V. Therefore it is ounded (say) y M as t and V. Now property (ii) for all ( for ε 2M I 2 < M ε 2M = ε 2 V t g e V (t) = β 1 β 2 also ) there exists M such that for V >, t > g e V t β 1 β 2 < ε 2M (5.9) Using (5.8) and (5.9) statement (5.7) ecomes s v+1 y η +1 F s, y β s Γ(v + 1)B(m, η, m, k, q, p) < ε Hence the theorem. References: [1]. Bhosale, B. N. and Chaudhary, M. S.: Fourier-HankelTtansform of Distriution of compact support, J. Indian Acad. Math, 24(1), 22, 169-19. [2]. Sharma, V. D and Gudadhe, A. S.: Some Operators On Distriutional Fourier-Mellin Transform, Science J.GVISH, 2, 25, 38-42. [3]. Kene, R. V. and Gudadhe, A. S.: On Distriutional Generalized Mellin-Whittaker Transform, Aryahata Research Journal of Physical Sciences, 12(5), 29, 45-48. [4]. Kene, R. V. and Gudadhe, A. S.: Some Properties of Generalized Mellin-Whittaker transform, Int. J. Contemp. Math. Sciences, 7(1), 212, 477-488. [5]. Zemanian, A. H.: Distriution Theory and Transform Analysis(McGraw-Hill, New ork, 1965). 23 Page