K [f(t)] 2 [ (st) /2 K A GENERALIZED MEIJER TRANSFORMATION. Ku(z) ()x) t -)-I e. K(z) r( + ) () (t 2 I) -1/2 e -zt dt, G. L. N. RAO L.
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1 Iterat. J. Math. & Math. Scl. Vl. 8 N. 2 (1985) A GENERALIZED MEIJER TRANSFORMATION G. L. N. RAO Departmet f Mathematics Jamshedpur C-perative Cllege f the Rachi Uiversity Jamshedpur, Idia ad L. DEBNATH Departmet f Mathematics Ulvreslty f Cetral Flrida Orlad, Flrida 32816, U.S.A. (Received May 13, 1983 ad i revised frm Jauary 15, 1985) ABSTRACT. I a series f papers [I-6], Kratzel studies a geeralized versi f the classical Meljer trasfrmati with the Kerel fucti (st) (q, + I; (st)q). This trasfrmati is referred t as GM trasfrmati which reduces t the classical Meijer trasfrm whe q I. He als discussed a secd geeralizati f the Meijer trasfrm ivlvig the Kerel fucti ()(x) which reduces t the Meijer fucti whe 2 ad the Laplace trasfrm whe I. This is called the Meijer-Laplace (r ML) trasfrmati. This paper is ccered with a study f bth GM ad ML trasfrms i the distributial sese. Several prperties f these trasfrmatis icludig iversi, uiqueess, ad aalytlclty are discussed i sme detail. KEY WORDS AND PHRASES. Distributial GM ad trasfrms, Meijer Trasfrm AMS MATHEMATICS SUBJECT CLASSIFICATION COD,S. 46F12, 44A20. I. INTRODUCTION. itegral I Zemala s bk [7, plt0] the Meijer trasfrmati is defied by meas f the K [f(t)] 2 [ (st) /2 K (2 s/) f(t) dt, (I.I) where K (z) is the mdified Bessel fucti f third kid f rder, ad has the itegral represetati [7, p148] K(z) r( + ) () (t 2 I) -1/2 e -zt dt, (1.2) fr Re -I, Re z 0. A alterative frm f (1.2) is Ku(z) ()x) t -)-I e z 2 4t dt
2 _ 360 G. L. N. RAO AND L. DEBNATH Kratzel [I, p149] has itrduced a geeralizati f the Meijer trasfrmati i the frm F(s) K (q) f(t)} t- =I (st) (q s + I; (st) q f(t) at, (1.4) where q t ad larg s (I + ). I his ther paper [3, p143], Kratzel csidered a itegral represetati f (O, B; z) i the frm (P B;. z) e -t zt-o at (1.5) where p > ad larc z < Whe p I, B + I, z 2 z 2 (l, + I; -)= 2(-) /2 K(z) (1.6) Result (1.4) reduces t (I.I) whe q i. Als, Kratzel itrduced a secd geeralizati f the Meijer trasfrmati ([!, p148], [, p328], [, p 369], [!, p 383] ad [, plod]) i the frm F(s) e () {f(t)} = ) {(st) I/} f(t) dt (1.7) where Re v >! Re {(st) i} > 0 ad the Kerel kv()(z) is give by -I -- z) ()(z) (2) -2-- ( r (+ I-) with Re I, Re z > 0 ad 1,2,3 t -zt I) e dt, (1.8) It is ted that (1.7) reduces t (i.i) whe 2, ad t the Laplace trasfrm whe I. Als, (1.7) is a special case f a mre geeral trasfrmati studied by Dimvski [, p23;, p141; I_0, p156]. The purpse f this paper is t study bth (1.4) ad (1.7) i the distributial sese ad establish therems ccerig cmplex iversi, uiqueess ad aalyticity. 2. DIFFERENTIAL OPERATORS. we use the tati ad the termilgy similar t thse f Kratzel [! ] ad Zemaia [7, pplto-200]. The fllwig differetial peratrs will be eeded fr this study: S,q (t) [t -q+l Dt{tq- Dtq (t)}] k, k O, I, 2 (2.1) where (t) is a cmplex smth fucti. M,[%() (t)] Dt-I t [t l- D t % ()(t)], I, 2,. (2 2) where l()(t) is defied i (1.8). The peratrs (2.1) ad (2.2) will be used t i=tlgae (1.4) ad (1.7) respectively.
3 GENERALIZED MEIJER TRANSFORMATION FUNCTION SPACE K, a AND ITS DUAL. We defie the fllwig semirms certai cmplex smth fuctis (t) (Zemaia [7, p176]): k,a () Sup le at tv- S,q (t) (3.1) < t < where a is a real umber, v is a cmplex umber with Re v. We ext defie Kv, a as the liear space f all fuctis (t) < t fr which the semirms y exist fr each k 0, I, 2, Each y is a semirm,a,a which is cmplete ad hece a Frechet space. We te that D(1) is subspace f The differetial peratr S,q is a ctiuus liear mappig f Kv, a it it- Kv, a Kv,aself [7, p171]. It is ted that the differetial peratr is slightly differet frm that used i the bk ]. LEMMA 3.1: (z) z v (q, v + I; z q) (3.2) where Re v ad q _> I, larg z < (I + ), the (st) E Kv,a fr every t i (,) ad fr every fixed zer s. PROOF: We have frm (3.1) k v,a(st) Sup feat t-1/2 k Sv, q O < t < Makig referece t [I, p153], we use the fact (st) Re v > 0. S k v,q (st) (-I) k(q+l) s k (st) (3.3) cmbied with the asympttic prperty f (t) [I, p 153] as t O. We prve that, as t 0 the semirms k v,a (st) are fiite fr v ad fr every fixed s O. Als, as t, it ca be shw that y are fiite fr a 0 which fllws m the,a asympttic prperty f the fucti [1, p 149]. DEFINITION i: The distributial geeralized Meijer trasfrm f f(t) is defied by F(s) K q) f(t) <f(t) (st) q (q + I; (st) q (3 4),a fr every s i f {s; s # 0, larg sl (I + I ad q t I}, where < f, represets the umber assiged t sme i a testig fucti space by a member f the dual space. I shrt, we call it as the distributial GM trasfrm f f. Sice by Lemma 3.1, (st) E Kv, a fr every fixed zer s, ad fr v > ; defiiti (3.4) has a sese as the applicati f f(t) E K, a t (st) E Kv, a where a is ay egative real umber ad Kv, a is the dual space f Kv, a- DEFINITION 2: A distributi f is called a GM-trasfrmable distributi if f Kv, a fr sme real umber a. NOTE: Lemma 3.1 is t true fr (i) Re v O, v # 0; (ii) v O; ad (ill) Re v < ANALYTIClTY OF F (s) The aalyticity f F(s) ca be expressed i the fllwig therem: THEOREM 4.1: F(s) f(t), (st) q (q, v + I; (st) q (4.1) fr s 9f, the F(s) is aalytic 9f; ad D s F(s) f(t), D s (st) q (q, v + I; (st) q (4.2)
4 362 G. L. N. RAO AND L. DEBNATH PROOF: A fairly stadard prcedure ca be used t prve this therem. Hwever, we state sme iitial steps fr the prf. where F (s + As) -F(s) < f(t) D s (st)q (q, + I; (st) q > < f(t) As(t) > As (4.3) 1?As s (t) [(st + Ast) (q,9 + I; (st + Ast) q (st) q (q,9 + I; (st)q)] We use the series expasi f frm [6, p 142] as (q,; z) (i q) (-z) + E (-----) = m= (4.4) m-+.-l-m. m q (-I) z (4.5) ad the asympttic behavir f as give i [, p 142]. After sme calculati, it ca be shw that D s (st) q (q,v + I; (st) q e Kv, a (4.6) s that (4.2) ad (4.3) have a sese. We ext fllw the argumets give i [7, pp ] cmbied with the use f Cauchy s itegral frmula t cmplete the prf f the therem. 5. FUNCTION SPACE Gv, a We w defie Gv, a AND ITS DUAL as the liear space f all cmplex-valued smth fuctis #(t) < t =. The tplgy f this space is geerated by a set f semirms Or,a, as v,a, () (t) where M, <SP< le at t- Mv, %)(t)l, (5.1) is the differetial peratr defied by (2.2). It is ted that (5.1) exists. We dete the dual space f Gv, a by Gv, a- LEMMA 5.1: #(st) ^. fr Re, the #(st) e Gg, a prvided () {v(st)n} (5.2) PROOF: It fllws frm [3, p 371] that fr t i (,=)ad fr every fixed s such that s # -I d M k %()(z) z l-v d 9, [z ) (z)] (-I) z A.()(z), (k=) dz-i dz "()(z) give i [3, p 371] i the frm Usig the fllwig asympttic prperty f A9 -I () (z) F( + ) + 0(I) as z, Re, (5.3) we btai % v,,a v {(st) I/} Sup Je at t -I (st) I/ %) {(st)i/}l <t< which are fiite fr each =l,2 as t if.() sff {(st) I/} O < t are fiite prvided > We ext csider the case fr t =. Fr <arg z<, z (st) I/ we use Sup le at t-- equati (7) f [3, p 372] t btai
5 () e at t v-1/2+ E se ^v GENERALIZED MEIJER TRANSFORMATION 363 ((st)l/} -1 e at tv-1/2- se (2) {(st)l/} v(-1) + E- - - This expressi is asympttically equal t -I {(-l) + I)E e at s t v-+ s (2)--2-- {(-l) v + I} xt e_(st)e,s# which is fiite if a <. REMARK: Eve if we take a mre geeral differetial peratr (that is, f a greater rder, say k) it must ivlve terms exp[-(st)i/sympttlcally as t, which teds t zer as t. DEFINITION 3: A distributi f(t) is called a M-L trasfrmable distributi if f(t) G, a fr sme real umber a ad Re I. DEFINITION 4: The M-L trasfrm f a M-L trasfrmable distributi g a G is defied by G(s) g(t), k)t {(st)} > (5.4) where s fl {s, Re s ; < arg s } which is give i [, p 372]. 6. COMPLEX INVERSION THEOREM FOR THE TRANSFORM (1.4). sese. Kratzel [, p 151] prved a iversi therem fr (, p 151) i the classical I rder t discuss a cmplex iversi therem, we eed the Wright fucti q, a; z) defied i [II] i the frm z (q, =; z) E! F(q + ) (6.1) = This reduces t Bessel fucti fr q I, that is, z 2 (I, + I;--g-) ()-" J(z) (6.2) Oe f the prperties f the Wright fucti [I, p 151] ca be expressed as v + v q t)} q s-1 " which is eeded i prvig the fllwig therem: THEOREM 6.1: (1) G(s) is hlmrphic i where (6.3) a {s; Re s c, [arg s[ 2 (1 + 1, q g 1, (ii) g(t) -v G() 0 ---; (t)) d, (6.4) (6.5) where the path f itegrati L is give by the (s) K q) {g(t)} (6.6)
6 364 G. L. N. RAO AND L. DEBNATH I ther wrds, we prve that, fr ay (s) E D (I) i the sese f cvergece i D (I: K(q){g(t)}, (s) G(s), (s) (6.7) where K q) is give by (1.4) PROOF: I view f cditi (ii) f the therem, the left had side f (6.7) ca be writte as 2i G() ( ---; (t)) d, (st) (q, + I; (st) q >, 0(s) > q_ +I (s 2i (q + I; (st) q 0 t, ---; (t)) dt G () d, O(s) > I (say) I view f (6.3), this expressi yields G() I d (s) > 2i s (6.8) which is equal t, usig a relati i [_I, p 152] G(s), (s) > This cmpletes the prf. We shall give here a weaker versi f a uiqueess therem. THEOREM 6.2: ad F(s) K (q) f(t) f G(s) K (q) g(t) F(s) G(s) Zf (h g, the f(t) g(t) i the sese f equality i D (1). PROOF: By Iversi Therem (6.1), we have F(s) G(s) K q)- [f(t)]-k q) [g(t)] This implies that f(t) g(t) i f g g K q) (t) g(t)] 0 i tip g. i the sese f equality i D (1). 7. CLOSING REMARKS: A trasfrm mre geeral tha (1.4) ad (1.7) was itrduced by Oberchkff i A mdified versi f that trasfrm was studied by Dimrskl [ IO] wh prved bth real ad cmplex iversi therems. We wuld like t discuss sme f these therems i the sese f distributi i a subsequet paper. ACKNOWLEDGEMENT: The first authr expresses his grateful thaks t Prfessrs E. Kratzel ad I. H. Dimvski fr their help. Thaks are due t Prfessr H. J. Glaeske fr his kid ivitati ad hspitality t the first authr at Jea. Authrs wuld live t express their thaks t Prfessrs Kratzel ad Glaeske fr useful discussis the subject f matter f this paper.
7 GENERALIZED MEIJER TRANSFORMATION 365 REFERENCES I. KRATZEL, E. Itegral Trasfrmati f Bessel-type, Prceedls f iteratial Cferece Geeralized Fuctis ad Operatial Calculus, Vara, (1975) KRATZEL, E., Bemerkuger Zur Meljer-Trasfrmatl ud Aweduge, Math Nachr. 30 (1965) KRATZEL, E., Eie Verallgemeierug der Laplace ud Meljer Trasfrmati, Wiss. Z. Uiv. Jea. Math Naturw Reihe, Heft 5 (1965) KRATZEL, E. Die Faltug der L-Trasfrmati, Wiss. Reihe, Heft 5 (1965) Uiv. Jea, Math Naturw. 5. KRATZEL, E. Differetiatis Satze der L-Trasfrmatl ud Differetial gleichuge ach dem peratr, 2 d te I- t ], d- (t )-I t+l Math Nachr 35 (1967) KRATZEL, E. ad MENZER, H., Verallgereierte Hakel Fuctie, Pub. Math. Debrece, 18, Fasc I-4 (1971) ZEMANIAN, A.H. Geeralized Itegral Trasfrmati, Itersciece, New yrk (1968). 8. DIMOVSKI, I. H. O a Bessel-Type Itegral Trasfrmati due t Obrechkff, Cmpt. Red. Acad. Bulg. Sci. 2_7 (1974) DIMOVSKI, I. H. A Trasfrm Apprach t Operatial Calculus fr the Geeral Besseltype Differetial Operatr, Cmpt. Red. Acad. Bu_. Scl. 27 (1974) I0. DIMOVSKI, I. H. O a Itegral Trasfrmati due t Obrechkff, Prc. f the Cferece Aalytic Fuctis, Kzublk, Lecture Ntes, 798 (1979) , Spriger Verlag. 11. WRIGHT, E.M. The Asympttic Expasi f Geeralized Bessel Fucti, Prc. Ld. Math. Sc. 2 (1934)
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