Abelian Theorem for Generalized Fourier-Laplace Transform

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1 Iteratioal Joural o Iovatio ad Applied Studies ISSN Vol. 8 No. 2 Sep. 204, pp Iovative Space o Scietiic Research Jourals Abelia Theore or Geeralized Fourier-Laplace Trasor V. D. Shara ad A. N. Ragari 2 Departet o Matheatics, Arts, Coerce ad Sciece College, Aravati, Aravati, Maharashtra, Idia 2 Departet o Matheatics, Adarsh College, Dhaagao Rly., Aravati, Maharashtra, Idia Copyright 204 ISSR Jourals. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. ABSTRACT: Itegral trasoratios have bee successully used or alost two ceturies i solvig ay probles i applied atheatics, atheatical physics, egieerig ad techology. The origi o the itegral trasors is the Fourier ad Laplace trasors. Itegral trasoratio is oe o the well kow techiques used or uctio trasoratio. Itegral trasor ethods have proved to be o great iportace i iitial ad boudary value probles o partial dieretial equatio. The Fourier as well as Laplace trasors have various applicatios i various ields like sciece, physics, atheatics, egieerig, geophysics, edical, cellular phoes, electroics ad ay ore as we have see i earlier papers. I this paper we discussed about Fourier-Laplace trasors ad this Fourier-Laplace trasor ay also have ay applicatios i various ields. This paper presets the geeralizatio o Fourier-Laplace trasor i distributioal sese. Testig uctio spaces are deied, also soe Abelia theores o the iitial ad ial value type are give. Ad the Abelia theores are o cosiderable iportace i solvig boudary value probles o atheatical physics. KEYWORDS: Fourier trasor, Laplace trasor, Fourier- Laplace trasor, geeralized uctio, Testig uctio space. INTRODUCTION Trasor ethods are widely used i ay areas o sciece ad egieerig. For exaple, trasor ethods are used i sigal processig ad circuit aalysis, i applicatios o probability theory. The theory o itegral trasor has preseted a direct ad systeatic techique or the resolutio o certai type o classical boudary ad iitial value probles []. The origi o the itegral trasors icludig the Laplace ad Fourier trasors ca be traced back to celebrated work o P. S. Laplace i o probability theory i the 780s ad to ouetal treatise o Joseph Fourier ( ) o La Theorie Aalytique de la Chaleur published i 822 [2]. May liear boudary value ad iitial value probles i applied atheatics, atheatical physics ad egieerig sciece ca be eectively solved by the use o Fourier trasor. Fourier ad Laplace trasors have used i Mechaical etworks cosistig o sprigs, asses ad dapers, or the productio o shock absorbers or exaple, processes to aalyze cheical copoets, optical systes, ad coputer progras to process digitized souds or iages, ca all be cosidered as systes[3]. The Fourier as well as Laplace trasors have various applicatios i various ields like sciece, physics, atheatics, egieerig, geophysics, edical, cellular phoes, electroics ad ay ore as we have see i earlier papers. I this paper we prove the Abelia theores. The settig or these Abelia theores is otivated by the iitial ad ial value results o Doetsch [4]. I atheatical aalysis, the iitial value theore is a theore used to relate requecy doai expressio to the tie doai behavior as tie approaches zero. Ad the ial value theore (FVT) is oe o several siilar Correspodig Author: V. D. Shara 549

2 Abelia Theore or Geeralized Fourier-Laplace Trasor theores used to relate requecy doai expressio to the tie doai behavior as tie approaches iiity. A ial value theore allows the tie doai behavior to be directly calculated by takig a it o a requecy doai expressio, as opposed to covertig to a tie doai expressio ad takig its it. The Abelia theores are obtaied as the coplex variable o the trasor approaches 0or i absolute value iside a wedge regio i the right hal plae. I the preset paper we established the Abelia theore or geeralized Fourier-Laplace trasor. I sectio 2, we have deied soe testig uctio spaces. We have give the deiitios i sectio 3. Sectio 4 is devoted the deiitio o distributioal geeralized Fourier-Laplace trasor. Iitial value theore or geeralized Fourier-Laplace trasor is give i sectio 5. Fially i sectio 6, ial value theore or geeralized Fourier-Laplace trasor is established. Lastly coclusios are give i sectio 7. Notatios ad teriology as per Zeaia [5],[6]. 2 TESTING FUNCTION SPACES 2. THE SPACE, This space is give by Sup FL E t x t t e D D t x CA k B l 0 < x < β k l q k kα l lβ a, α = φ : φ / ρa, k, q, lφ(, ) = 0 < < t xφ(, ) Where, k, l, q = 0,, 2,3,..., ad the costats A, Bdepeds o the testig uctio φ. (2.) 2.2 THE SPACE, It is give by Sup FL E t x t t e D D t x C A q 0 < x < k l q q qγ a, γ = φ : φ / ξa, k, q, lφ(, ) = 0 < < t xφ(, ) lk Where, k, l, q = 0,, 2,3,..., ad the costats deped o the testig uctio φ. (2.2) 3 DEFINITIONS The Fourier trasor with paraeter s o ( ) [ ] ist t deoted by F [ t ] ( ) = F( s) ad is give by F ( t) = F( s) = e ( t) dt, or paraeter s > 0. (3.) The Laplace trasor with paraeter p o ( ) [ ] 0 px x deoted by L[ x ] ( ) F( p) = ad is give by L ( x) = F( p) = e ( x) dx, or paraeter p > 0. (3.2) The Covetioal Fourier-Laplace trasor is deied as { } 0 FL ( t, x) F( s, p) ( t, x) K( t, x) dtdx Where, = =, (3.3) i( st ipx) K( t, x) = e 4 DISTRIBUTIONAL GENERALIZED FOURIER-LAPLACE TRANSFORMS (FLT) For ( t, x) FL β a, α, where FL β a, α is the dual space o FL β a, α. It cotais all distributios o copact support. The distributioal Fourier-Laplace trasor is a uctio o ( t, x) ad is deied as ISSN : Vol. 8 No. 2, Sep

3 V. D. Shara ad A. N. Ragari { } i( st ipx) = =, (4.) FL ( t, x) F( s, p) ( t, x), e Where, or each ixed t ( 0 < t < ), x ( 0 x ) applicatio o ( t, x) FL β i( st ipx) a, α to e FL β a, α. < <, s > 0ad p > 0, the right had side o (4.) has a sese as a 5 INITIAL VALUE THEOREM FOR GENERALIZED FOURIER-LAPLACE TRANSFORM CONDITIONS: (i) ( t, x ) = 0 or < t < T, 0 < x < (ii) There exist real ubers sad p such that 0 < x <. i( st ipx) ( t, x) e is absolutely itegrable over < t <, 5. THEOREM I the locally itegrable uctio ( t, x ) satisyig above coditios with T = 0 ad i there exist a coplex uber A ad a real uber ad such that (i) > (ii) > ad Γ ( ) ( t, x) t, x 0 ct x (5..) The: s ( is) p F( s, p), where c = Γ ( ) PROOF: By Zeaia [5] pp.243 ad Debath Lokeath [2] pp. 36, we kow that (i) (ii) ist Γ ( ) t e dt =, or ( is) px Γ ( ) x e dx =, or p 0 > ad s > 0 ad >, p > 0 so that ( is) p F( s, p) cat x ist px = ( is) p ( t, x) e e dtdx Γ ( ) T 0 0 ( is) p { ( t, x) Γ ( ) X 0 ( t, x) Γ ( ) T ( t, x) Γ ( ) X ( t, x) } Γ ( ) ISSN : Vol. 8 No. 2, Sep

4 Abelia Theore or Geeralized Fourier-Laplace Trasor By Zeaia [5] pp. 243 ad D. V. Widder [] pp.8 or ay positive ε we ca id a costat M such that ist At M ( is) i( s ε ) T ( is) e α( t) dt < e or s > ε Γ ( ) i( s ε ) T Where right had side o this iequality is approaches zero as s becoes iiite ad by D. V. Widder [] pp.8 we have px Ax Mp ( p ε ) X p e α( x) dx < e or p > ε, thereore Γ ( ) p ε X s ( is) p F( s, p) Sice T ad X are arbitrary Fro which the result ollows. ct x ( t, x) Γ ( ) i( st ipx) s ( is) p e dtdx Γ ( ) ct x 0 Sup ( t, x) Γ ( ) t T ct x 0 x X ( t, x) Γ ( ) t 0 ct x x LEMMA I ( t, x) FL β a, α with its support i t F( s, p) Mt e k, where M is suicietly large costat. t ad x x where t > 0ad x > 0 the PROOF: Let g( t, x) be a sooth uctio o 0 t ad 0 x such that g( t, x ) = o t, ) g( t, x ) = 0o ( 0,T ) ad ( 0, X ) where T < t ad X x ad x, ) ad <. As a distributio o slow growth satisies a boudedess property o distributio, there exists a positive costat K ad a o-egative iteger ρ such that l q Where, g ( t, x) Sup k l q F( s, p) K t e Dt Dx g( t, x) ( t, x) 0 t ρ 0 t, x < gives k l q K sup t e g ( t, x) ( t, x) th l derivatives o g( t, x) with respect to t ad sup k K t l e th q derivative o g( t, x) with respect to x. ISSN : Vol. 8 No. 2, Sep

5 V. D. Shara ad A. N. Ragari FL β α As ( t, x) is a eber o a, ad k, l = 0,,2,... K supt k e k Mt e Where M is suicietly large costat. 5.3 THEOREM I ( t, x) is decoposed ito ( t, x) = ( t, x) 2( t, x) where ( t, x ) is the ordiary FL trasorable uctio satisyig the hypothesis o theore 5. ad 2 ( t, x ) is a regular distributio satisyig the hypothesis o above lea, the s ( is) p F( s, p) PROOF: It ollows ro above lea that (i) s ( is) p F( s, p) = 0 Ad ro theore 5., we have (ii) s ( is) p F( s, p) that 2 Moreover the distributioal geeralized FL trasor F o equals the ordiary geeralized FL trasor o, so F( s, p) = F ( s, p) F ( s, p), thereore (i) ad (ii) proves theore. 6 SOME ABELIAN THEOREMS OF THE FINAL VALUE TYPE 6. THEOREM For a locally itegrable uctio ( t, x ) satisyig coditios with T = 0 Γ ( ) ( t, x) a real uber ad such that (i) >,(ii) > ad t, x ct x s 0 ( is) p F( s, p), where c = Γ ( ) p ad existece o ay coplex costat Aad, the ISSN : Vol. 8 No. 2, Sep

6 Abelia Theore or Geeralized Fourier-Laplace Trasor PROOF: We proceed as i theore 5. to obtai ( is) p F( s, p) T 0 ( is) p { ( t, x) Γ ( ) X 0 ( t, x) Γ ( ) T ( t, x) Γ ( ) X ( t, x) } Γ ( ) Thereore, s 0 ( is) p F( s, p) p cat x Γ ( ) i( st ipx) Γ ( ) cat x s 0 ( is) p ( t, x) e dtdx p Γ ( ) sup ( t, x) Sice T ad X are arbitrary. cat x Γ ( ) t, x cat x ( t, x) Now the proo ca be easily copleted. 7 CONCLUSION The Fourier-Laplace Trasor is developed ad geeralized i the distributioal sese i this paper. Soe testig uctio spaces are give. This paper is aily ocuses o Abelia theores o Iitial value type ad Fial value type. REFERENCES [] S. Pooia ad R. Mathur, Coparative Study o Double Melli-Laplace ad Fourier Trasor, Iteratioal Joural o Matheatics Research, vol. 5, o., pp. 4-48, 203. [2] L. Debath ad D. Bhatta, Itegral Trasors ad their Applicatios, Chapa ad Hall/CRC Taylor ad Fracis Group Boca Rato Lodo, New York, [3] R. J. Beereds, H. G. ter Morsche, J. C. va de Berg ad E. M. va de Vrie, Fourier ad Laplace Trasors, Cabridge Uiversity Press, [4] G. Doetsch, Itroductio to the Theory ad Applicatio o the Laplace Trasoratio, Spriger-Verlag, New York, 974. [5] A.H. Zeaia, Distributio theory ad trasor aalysis, McGraw-Hill, New York, 965. [6] A. H. Zeaia, Geeralized itegral trasor, Iter sciece publisher, New York, 968. [7] V. D. Shara ad A. N. Ragari, Aalyticity o distributioal Fourier-Laplace trasor, Iteratioal J. o ath. Sci. ad Egg.Appls, vol. 5, o.v, pp , 20. [8] V. D. Shara ad A. N. Ragari, Geeralized Fourier-Laplace Trasor ad Its Aalytical Structure, Iteratioal Joural o Applied Matheatics ad Mechaics, vol. 3, o., pp. 4-46, 204. ISSN : Vol. 8 No. 2, Sep

7 V. D. Shara ad A. N. Ragari [9] V. D. Shara ad A.N. Ragari, Geeralizatio o Fourier-Laplace trasors, Archives o Applied Sciece Research, vol. 4, o. 6, pp , 202. [0] P. A. Gulhae ad A. S. Gudadhe, Boudary Value Proble or the distributioal LS geeralized LS trasor, ARJPS, vol. 8, o. -2, [] D. V. Widder, The Laplace Trasor, Priceto Ui press 955. [2] A. S. Gudadhe ad R. V. Kee, Abelia Theore or Geeralized Melli-Whittaker Trasor, IOSR Joural o Matheatics, vol., o. 4, pp. 9-23, 202. [3] A. Gupta, Fourier Trasor ad Its Applicatio i Cell Phoes, Iteratioal Joural o Scietiic ad Research Publicatios, vol. 3, o., pp [4] I. M. Gelad ad G. E. Shilov, Geeralized uctio, vol.ii, Acadeic Press, New York, 968. [5] R. D. Carichael ad R. S. Pathak, Abelia Theores or Whittaker Trasors, Iteratioal Joural o Math. & Math. Sci., vol. 0, o. 3 pp , 987. ISSN : Vol. 8 No. 2, Sep