f err (AZ)g(Z)dZ n (A),A 0; da=i<_jh dlj (nljxlj); nlj Sm, ^ E Sm; the space of m m real symmetric matrices parameterlzed by (j), rm<) (2i

Transcription

1 Intenat. J. Math. & Math. Sci. Vol. i0, No.3 (1987) GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES R.M. JOSHI and J.M.C. JOSHI Depatent of Matheatics Govt. P. G. College Pithoagah (U.P.) INDIA (Received Septebe 9, 1985) ABSTRACT. In the pesent pape we have extended genealized Laplace tansfos of Joshl to the space of syetic atices using the confluent hypegeoetlc function of atix aguent defined by Hez as kenel. Ou extension is given by g(z) <) (8 flfl( :8:- ^Z) f(^)d^ A>0 The convegence of this integal unde vaious conditions has also been discussed. The eal and coplex invesion theoes fo the tansfo have been poved and it has also been established that Hankel tansfo of functions of atix aguent ae liiting cases of the genealized Laplace tansfos. EY WORDS AND PHRASES. Integal tansfos, Laplace tansfo, Hankel tansfo of Hez, functions of atix aguent. 29B0 AMS SUBJECT CASSIFICATION CODES. 44F2. I. INTRODUCTION. A function of atix aguent is a eal o coplex valued function of the eleents of a atix. Let A be a syetic atix of diension The function f(^) is called syetic function if f(^) f(oao ), whee 0 E 0 the goup of othogonal atices. If f(^) is a syetic function, then it is a function of the eleentay functions llke tace, deteinant, etc., of A. Hez [I] has defined the Laplace tansfo with atix vaiables by g(z) et(-^z) f(^)d^, (I.I) whee ^ E S; the space of eal syetic atices paaetelzed by (j), Z X + iy, X,Y S * the space of eal syetic atices paaetelzed by X (nljxlj); nlj being if i j, 1/2 othewise. Also Lebesgue easue in S, is da=i<_jh dlj the tace e(a) stands fo e ^ The integation in (I. I) is ove the set of all positive definite ^ The invese tansfo (I.I) is given by (A),A 0; f e (AZ)g(Z)dZ n (2i./ Re(Z) X > 0 0, othewise o ( + I) 2 (I.2)

2 504 R.M. JOSHI AND J.M.C. JOSHI Hez [13 has used (I.I) and (1.2) in defining hypegeoetic functions of atix aguent. The confluent hypegeoetic function, IF1 is defined by (b) IFl(a; b; M) fet(z) det(e- MZ) -a (det Z) -b dz, (1.3) (2i) n Re(Z) X > 0 o which holds fo abitay coplex M and a Re(b) >, povided we take X > Re(M) Hee (b) is the genealized gaa function of Siegel, and p ( + I)/2. Hez [I] has shown that (b) ifl(a; b; Z)= IEet(ZR) (det R) a-p det(e-r)b-a-pdr, (1.4) (a)(b_a) 0 which holds fo Re(a), Re(b) Re(b-a) > p-l. In the pesent pape, we shall extend the genealized Laplace tansfo of Joshi [2], to the space of eal syetic atices. The kenel will be IF1 function of atix aguent. We shall also pove soe theoes on the popeties of the tansfo thus defined. In section 3, we shall establish a elation between Laplace tansfo and genealized Laplace tansfo though the opeato of factional integation defined by ading 3 I a f(^) -l,a) I^f(R) det(^-r) a-p dr, (1.5) 0 holding unde suitable conditions on the function f(r). Integal (1.5) is a genealization of Rieann Liouville integal to atix vaiables. 2. DEFINITION. Joshi [2] has defined the genealized Laplace tansfo in the scala case by g(x) (a+b+c+l)(b+c+l) 0 (xy) b IF (b+c+l a+b+c+l -xy) f (y) dy (2 I) In analogy with (I.I), we conside the elation (a) g(z) -- f^ >0 IF1 (a; b;-^z) f(^) d^; z x + iy, (2.2) whee ^ S, X, y S* and F1 is the function defined by (1.3). If integal (2.2) conveges in soe Z egion, then it epesents a function, g(z), of the eleents of the coplex syetic atix Z thee. In this case, g(z) will be called genealized Laplace tansfo of f (^). Now we have to establish the convegence of (2.2) to find the egion whee it defines a function. But, befoe poving the convegence theoes, we have to say soething on the liit concept in atix space. We know that the space $ is a patially odeed set fo the ode elatio ">" defined in it by A > B <--> A-B is positive definite, whee A, B S S is also a diected set since fo any pai of atices A,B S thee exists a nube k such that A < ke and B < ke, E being unit atix. Fo any two functions f(^) and g(^) defined in S we will say that f (A) is of the ode of g(^) fo infinitely lage ^ (in the sense of the ode elation; >), and will wite f(^) g(a); A (),

3 GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES 505 if Mooe-Slth liit (Yoshida [5], p. 103) of f(^)/g(^) though the diected set S is equal to I. 3. CONVERGENCE THEOREMS. Befoe poving the theoes on convegence of the integal (2.2), we will pove a few leas which will be useful in poving convegence theoes. LEMMA I(A) If Re(b) >, and Re(b-a) > p-l, Fo(b) -a IFI (a; b; -M) (det M) M (=). (b-a) (B) If Re(b) >, and Re(a) > p-l, (b) IF1 (a; b; M)~ et(m) (det M (a We have, fo (1.3), -(b-a) M (=). (3.2) 1" (b) -l)-a (det Z) dz Re(b) > ifl(a; b; -M) f et(z) det(e- Mz -b (2i) n Re(Z) =X (b) -a -(b-a) (2i) n (det M) /et(z) (det Z) det(e-zm- )-adz Re (Z) X Now, if M (), applying foula (I.I) of Hez [I], we obtain 1Fl(a; This poves pat (A) of the lea. (b) a b; -M) (det M) (b-a) Re(b-a) > p-1. To pove the pat (B), we use Kue s foula (Hez [1]) 1Fl(a; b; M) e(m) 1Fl(b-a; b; -M). Using pat (A) of the lea, and applying Kue s foula, pat (B) of the lea is poved. LEMMA 2. e(b) (6) ifl(a;,b; (a-) -AZ)(det ^)-P da (det Z) (3.3) >b (a) (b-6) whee Re(a), Re(b), Re(6), Re(b-a), Re(b-6), Re(a-6) > p-l, and Re(Z) > 0. Fo (1.4), we have (a) (b-a) (b) ifl(a; b; -Z) --fe et(-raz) (det R) a-p det (E-R) b-a-p dr; 0 whee fo Re(a), Re(b), Re(b-a) > p-l. Substituting the value of ifl(a, b, -AZ) in the left hand side of (3.3), and changing the ode of integation, we have if1(a; (b) ( b; -AZ)(det ^)-P d ^= (a)(b-a) 8(a-, >0 (a- b-a) is the genealized beta function of Siegel and whee 8 (a-6) (b-a) b-a) (det Z) 8 (a-,b-a) (b-) This poves the lea. Now we pove: THEOREM I. (a) If the integal (2.2) conveges absolutely fo Z Z 0 fo which < Re(Z0) 0, then it conveges absolutely fo evey Z fo which Re(Z) > 0 povided

4 506 R.M. JOSHI AND J.M.C. JOSHI Re(b) >, Re(b-a) > p-l. (b) If the integal (2.2) conveges absolutely fo Z Z 0 whee Re(Z0) < O, then it conveges absolutely fo evey Z fo Re(Z0) < Re(Z) < 0, povided Re(a) > p-l, Re(b) >. (c) If a b, and (2.2) conveges absolutley fo Z Z 0 then it conveges fo evey Z fo which Re(Z) > Re(Z0) PROOF. We have f f ^0 Now, as ^/() fo lea I, we have ilfl(a; b; -AZ) ifl(a; l(det Az)-al b; AZ 0) (detazo)-a IFl(a; b; -Z) llfl(a; b;-z)f(^)id^-- F (a; b;-z) ifl(a; b;-azo)f(^)d^ (3.4) l(det ZO Z-l)al povided we have Re(b) >, Re(b-a) > p-l, Re(Z), Re(Z0) > 0. But ifl(a; b; -^Z) is bounded in the set of positive definite ^ and does not vanish anywhee thee, povided Re(b) >, Re(b-a) > p-l, and Re(Z) > 0. Thus we see that ifl(a; b; -AZ) ifl(a; b; -AZ 0) is bounded in the set of p.d. ^, povided Re(b) >, Re(b-a) > p-l, and Re(Z) > O. f ll Fl(a; b; -AZ)E(^)Id^ < k.llfl(a; b; -AZ0)f(A)Id^ Now (3.4) educes to ^0 A>O whee k is the least uppe bound (1.u.b.) of ifl(a; b; -AZ) ifl(a; b; -AZ 0) This poves pat (a) of the theoe. Now, if Re(Z) < O, Re(ZO) < 0, Re(b) >, Re(a) > p-l, we have fo lea l(b), ifl(a; b; -AZ) b-a ifl(a; b; -AZ 0) let(-a(z-z0) )(det ZoZ-l) I; So, as in the poof of pat (a) of the theoe, we once again note that ifl(a; b; --AZ) ifl(a; b; -^Z 0) is a bounded function in the patially odeed set, ^ povided Re(b) >, Re(a), Re(b-a) > p-l. So, it follows fo inequality (1.4), that the integal (2.2) is absolutely convegent. This poves pat (b) of the theoe. We have yet to discuss the case a b. In this case, ifl(a; b; -^Z) is equal to et(-az), so that the tansfo (2.2) is educed to the Laplace tansfo (I.I) which is absolutley convegent fo all Z fo which Re(Z) > Re(Z0), povided the integal (2.2) is absolutely convegent fo Z Z O. This poves pat (c) of the theoe. THEOREM 2. If f(^) is absolutely integable in the patially odeed set ^ i.e.,

5 GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES 507 sl^l then the integal (2.2) is absolutely convegent fo all Z, fo which Re(Z) > 0, Re(a), Re(b), Re(b-a) > p-l. PROOF. Since ifl(a; b; -AZ) is bounded in the patially odeed set ^ povided Re(Z) > 0, Re(a), Re(b), Re(b-a) > p-l, theefoe we have /IIFl(a; b;-z)f(^)id^_< /IIFl(a; b;-^z)iif(^)id^_< k.iif(^)id^ A>O A>O A>O whee k is l.u.b, of ifl(a; b; -AZ) in the patially odeed set A fo Re(Z) > 0. This poves the theoe. THEOREM 3. If, fo soe nube d and a posltve nube k, If(^)l < el (et) d-p ;A > 0, then the integal (2.2) is absolutely convegent fo Re(Z) > 0, povided Re(a), Re(b), Re(b-a), Re(a-d), Re(b-d) > p-l. PROOF. We have, fo lea 2, IIIFI(a; b;-az)f6 )Id^_< k /llfl(a; b;-az)(det^ )d-pld^ < >0 ^ >0 whee Re(a), Re(b), Re(b-a), Re(a-d), Re(b-d) > p-l, and Re(Z) > 0. This poves the theoe. THEOREM 4. If, fo any eal syetic atix R, [f(^[ < k. et(-r^) [(det^)b-p[ then the integal (2.2) is absolutely convegent fo positive definite R povided Re(Z+R) > 0, Re(b) > p-l. PROOF. /lift(a; b; z)f(^)[d^ _< I[iF1(a; b;-^z)[[f(^)[d^ < <kfiifl(a; b; -^Z) et (-R^) (det^)b-p [d ^ <, A>O fo Hez ([i], foula (2.7)) povided we have Re(Z+R) > O, Re(R) > O, Re(b) > p-l. This poves the theoe. 4. GARDING S FRACTIONAL INTEGRAL AND GENERALIZED LAPLACE TRANSFORM. In this section, we have deived cetain connections between Laplace tansfo, Hankel tansfo, and genealized Laplace tanfo with the help of Gading s factional integal. Fo foula (1.4), we can show, by cetain change of vaiables, that (a) (4.1) 1F (a; b; -^z)(det Z) b-p fz et(_ra)(detr)a-pdet(z_r]--,ar (b) (b-a) o whee Re(a), Re(b), Re(b-a) > p-l. THEOREM 5. If (a) g(a; b; Z)= F ( ) f IFI (a; b;-^z)f(^) da (4.2) f et(-az)(det z)a-pf(h)da (4.3)

6 508 R.M. JOSHI AND J.M.C. JOSHI then I b-a (Z) (det z)d-p g(a; b; Z) (4.4) Povided integals (4.1) and (4.2) convege in the genealized ight half plane Re(Z)> and Re(a), Re(b-a) > p-l. PROOF. Applying the opeato of factional integation to both the sides of (4.3), we have by changing the ode of integation which is justified by Fubini s theoe I b-a q(z) 0( fz o Now, fo (4.1) we have in view of (4.2), F (b-a) ff ^)[ et(-r^)(det R) a-p det(z-r) b-a-p dr] d^ ^> I b-a (Z) (det Z) b-p g(a; b; Z). This poves the theoe. Fo (4.2) and (4.3) we note that (Z) g(a; a; Z)(det Z) a-p, since ifl(a; a; -AZ) et(-^z). Fo Gading s [3] theoy, if deivatives of a function, f(^) exist fo sufficiently high odes, D a i a f(^) f(^). Since (Z) in equation (4.3) is a Laplace tansfo, the deivatives of it will exist fo all odes. So fo (4.4), we have (Z) g(a; a; Z)(det Z) a-p Db-a[(det Z) b-p g(a; b; Z)]. (4.5) whee D b-a is the diffeential opeato of factional ode (Gading [3]). Moeove, fo ^ S D det(nij ), nij being if i j and 1/2 othewise And fo z D z det(3--ij). Foula (4.5) has been established fo eal Z, it follows fo all coplex Z fo which Re(Z) > 0 by analytic continuation. But we ust note that in any case (b-a) is a non negative intege geate than p-i and Re(a) > p-l. But the condition; (b-a) > p-i can be elaxed to the condition (b-a) > 0. To do so, we apply D-opeato to both the sides of (4.2) afte ultiplying it by (det z)b-p to get (4.5), but then we ust have; Re(a) >, and b-a should be a non negative intege. Laplace Foula (4.5) allows us to find a coplex invesion theoe fo the genealized tansfo. THEOREM 6. If, fo Re(a), Re(b) >, (a) F (b) "[ifl(a; b; -AZ)f(A)dA g(a; b; Z) ja > 0 is absolutely convegent in the genealized ight half plane given be Re(Z) > X 0 and S* et (X +iy) B-p g(a; b; X + iy)[dy < YS X> X o

7 then, fo A > 0, GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES 509 ll det (X+iY) b-p g(a; b; X+iY) dy 0 X/((R)) YeS* (-l)(b-a) F (b-a+p) (2i) nf (b) povided (b-a) is a non negative intege. 0 b; ^z) (det ^z)b-p g(a; b; Z)dZ f(^). PROOF Unde the conditions of the theoe, (det Z) b-p g(a; b; Z) can be epesented as a Laplace tansfo of a function, say (^), so that we have (det Z) b-p g(a; b; Z) et(-^z) (^) d^ (4.6) Now, applying D-opeato to both the sides of (4.6), we have 0 Db-a[ (det z)b-p > g(a; b; Z)] et(-^z)(det -^)b-a (^)d^ 0 Now, fo Hez([l]), foula (I.I)), (det Z) -a-p is the Laplace tansfo of (a-p) (det )a-2p povided Re(a) > 2p-I. So, fo the convolution popety of Laplace tansfo, (det Z) -a-p D b-a (det Z) b-p Laplace tansfo of g(a; b; Z)] is the f(^) la-p[det(-^) b-a (^)] (4.7) Now, inveting (4.6) by coplex Invesion foula of Laplace tansfo, (Hez [1]), which is peissible unde the conditions of the theoe, and then putting the values in (4.7), we obtain f(^) la-p[ fet (^Z) det (-^) b-a (det Z) b-p g(a; b; Z) dz] (2wilh Re(Z) X o Now, fo (4.1), we have (-I) e(b-a) (2i)n Re(Z) --X o f(^) (_l)(b-a) (b-a+p) (2i) n (b) This poves the theoe. fla-p[etz)(det ^)b-a](det z)b-p g(a; b; Z) dz lfl(b-a+p; b; ^Z)(det ^Z) b-p g(a; b; Z) dz Re(Z) X o COROLLARY. If the conditions of Theoe 6 hold, b-a 0 and then g(z) /et(-^z)f(^) d^ f (^) (p) F (b)(2i) n f#if1 Re(Z) X o (p; b; AZ)(det^ b-p Z) g(z)dz This coollay can be deduced by noting the siple fact that ifl(b; b; Z)= e(z). This coollay gives a new inveison foula fo the Laplace tansfo with atix (4.8)

8 510 R.M. JOSHI AND J.M.C. JOSHI vaiables. In the scala case; -- I, foula(4.8)educes to Roony s [4] foula. 5. HANKEL TRANSFORM AND GENERALIZED LAPLACE TRANSFORM. Hez [i] has defined Hankel tansfo of functions of atix aguent by g(t0 =/AC(^R)(det R)C f(r) dr (5.1) whee A is the Bessel function of atix aguent and f E L 2 L 2 being the C C C Hilbet space of functions fo which the no defined by --JR> 2 llfllc 0 If(R) 12 (dec R) c dr whee c is a eal nube geate than -I/2. We can find a connection between Hankel tansfo and genealized Laplace tansfo The esult can be stated in the fo of the following theoe. THEOREM 7. If f(^) L 2 and c-p F (a) g(z) (b) #fl F (a; b; -AZ) (deta)b-pf (^)d^ (5.2) coveges absolutely in the siply connected egion Re(a), Re(b) > p-i fo Z > 0, and then Li (a) g( z) 0(z) (5.4) a+ The poof of the theoe is quite siple in view of the liit (see Hez [I]): ei IFI (a; b; -R) (b) _p(r). a+ We shall now pove a theoe which seves as eal invesion theoe o the genealized Laplace tansfo. THEOREM 8. If (a) fl g(a; b; Z)= F(b Fl(a; b;-z)(det^) b-p f(^)d^ -A> 0 is absolutely convegent fo a > p-l, b > p-l, Z > 0, and f g ul2-p (5.5) then f( to ll a b a (a) H{g(a; b; ha) (5.6) whee H{g(a; b; az)} -P z>0 (aaz) (det Z) b-p g(a; b; Z)dZ. (5.7) PROOF. Since f e L it s Hankel tansfo will exist. So we have -P (Z) =f_p(az)(det^)b-pf(a)d^ Now, fo Theoe 7, we have (5.8)

9 GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES 511 (Z)--li "g(a; b;!z) (5.9) (a) a a By the Hankel invesion of (5.8) and (5.9), we obtain f(^) li a F A_p(AZ)(det Z. B-p g(a; b; Z) (5.10) (a) Now, changing vaiables fo Z to az, and noting that the Jacobian of tansfoation fo the change is (a) p we have the desied esult. REFERENCES i. HERZ, C. S., Bessel functions of atix aguent, Annals of Matheatics, 61(1955), JOSHI, J. M. C., Invesion and epesentation theoes fo a genealized Laplace tansfo, Pacific Jounal of Matheatics, 14(1965) GARDING, L., The solution of Cauchy s poble fo two totally hypebolic linea diffeential equations by eans of Riesz integals, Annals of Matheatics, 48 (1947) ROONY, P.G., A genealization of coplex invesion foula fo the Laplace tansfoation, Poc. Ae. Math. Soc., 5(1954) YOSHIDA, K., Functional Analysis, Naosa Publishing HOuse, New Delhi (1979).

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