Internat. J. Math. & Math. Sci. Vol. i0, No.3 (1987) 503-512 503 GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES R.M. JOSHI and J.M.C. JOSHI Department of Mathematics Govt. P. G. College Pithoragarh (U.P.) INDIA (Received September 9, 1985) ABSTRACT. In the present paper we have extended generalized Laplace transforms of Joshl to the space of m m symmetric matrices using the confluent hypergeometrlc function of matrix argument defined by Herz as kernel. Our extension is given by rm<) g(Z) r (8 flFl( :8:- ^Z) f(^)d^ m A>0 The convergence of this integral under various conditions has also been discussed. The real and complex inversion theorems for the transform have been proved and it has also been established that Hankel transform of functions of matrix argument are limiting cases of the generalized Laplace transforms. EY WORDS AND PHRASES. Integral transforms, Laplace transform, Hankel transform of Herz, functions of matrix argument. 29B0 AMS SUBJECT CASSIFICATION CODES. 44F2. I. INTRODUCTION. A function of matrix argument is a real or complex valued function of the elements of a matrix. Let A be a symmetric matrix of dimension m m The function f(^) is called symmetric function if f(^) f(oAo’), where 0 E 0 the group of m m m orthogonal matrices. If f(^) is a symmetric function, then it is a function of the elementary functions llke trace, determinant, etc., of A. Herz [I] has defined the Laplace transform with matrix variables by g(Z) etr(-^Z) f(^)d^, (I.I) ^>0 where ^ E Sm; the space of m m real symmetric matrices parameterlzed by (j), Z X + iY, X,Y S * m the space of m m real symmetric matrices parameterlzed by X (nljxlj); nlj being if i j, 1/2 otherwise. Also dA=i<_jH dlj is the trace ^ Lebesgue measure in Sm, err(A) stands for e The integration in (I. I) is over the set of all positive definite ^ The inverse transform (I.I) is given by (A),A 0; f err (AZ)g(Z)dZ -- n m(m + I) (2i ./ 2 Re(Z) X > 0 0, otherwise o (I .2)
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Internat. J. Math. & Math. Sci.Vol. i0, No.3 (1987) 503-512
503
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES
R.M. JOSHI and J.M.C. JOSHI
Department of MathematicsGovt. P. G. CollegePithoragarh (U.P.)
INDIA
(Received September 9, 1985)
ABSTRACT. In the present paper we have extended generalized Laplace transforms of Joshl
to the space of m m symmetric matrices using the confluent hypergeometrlc function
of matrix argument defined by Herz as kernel. Our extension is given by
rm<)g(Z) r (8 flFl( :8:- ^Z) f(^)d^
m A>0
The convergence of this integral under various conditions has also been discussed.
The real and complex inversion theorems for the transform have been proved and it has
also been established that Hankel transform of functions of matrix argument are limiting
cases of the generalized Laplace transforms.
EY WORDS AND PHRASES. Integral transforms, Laplace transform, Hankel transform of Herz,
functions of matrix argument.
29B0 AMS SUBJECT CASSIFICATION CODES. 44F2.
I. INTRODUCTION.
A function of matrix argument is a real or complex valued function of the elements
of a matrix. Let A be a symmetric matrix of dimension m m The function f(^) is
called symmetric function if f(^) f(oAo’), where 0 E 0 the group of m mm
orthogonal matrices. If f(^) is a symmetric function, then it is a function of the
elementary functions llke trace, determinant, etc., of A. Herz [I] has defined the
Laplace transform with matrix variables by
g(Z) etr(-^Z) f(^)d^, (I.I)^>0
where ^ E Sm; the space of m m real symmetric matrices parameterlzed by (j),Z X + iY, X,Y S*
mthe space of m m real symmetric matrices parameterlzed by
X (nljxlj); nlj being if i j, 1/2 otherwise. Also dA=i<_jH dlj is the
trace ^Lebesgue measure in Sm, err(A) stands for e The integration in (I. I) is over
the set of all positive definite ^ The inverse transform (I.I) is given by
(A),A 0;
f err (AZ)g(Z)dZ -- n m(m + I)
(2i ./2
Re(Z) X > 0 0, otherwiseo
(I .2)
504 R.M. JOSHI AND J.M.C. JOSHI
Herz [13 has used (I.I) and (1.2) in defining hypergeometric functions of matrix
argument. The confluent hypergeometric function, IF1 is defined by
rm(b)IFl(a; b; M) fetr(z) det(E- MZ) -a (det Z) -b dZ, (1.3)
(2i)n Re(Z) X > 0o
which holds for arbitrary complex M and a Re(b) > m, provided we take X > Re(M)
Here rm(b) is the generalized gamma function of Siegel, and p (m + I)/2.
COROLLARY. If the conditions of Theorem 6 hold, b-a 0 and
g(Z) /etr(-^Z)f(^) d^^>0
thenr (p)
f (^) mf
b-pF (b)(2i)n #IF1 (p; b; AZ)(det^ Z) g(Z)dZm Re(Z) X
o
This corollary can be deduced by noting the simple fact that
iFl(b; b; Z)= err(Z).
This corollary gives a new inverison fromula for the Laplace transform with matrix
(4.8)
510 R.M. JOSHI AND J.M.C. JOSHI
variables. In the scalar case; m-- I, formula(4.8)reduces to Roony’s [4] formula.
5. HANKEL TRANSFORM AND GENERALIZED LAPLACE TRANSFORM.
Herz [i] has defined Hankel transform of functions of matrix argument by
g(t0 =/AC(^R)(det R)C f(R) dR (5.1)
^>0where A is the Bessel function of matrix argument and f E L2 L2 being the
C C C
Hilbert space of functions for which the norm defined by
2 --JR> If(R) 12 (dec R) c dRllfllc0
where c is a real number greater than -I/2.
We can find a connection between Hankel transform and generalized Laplace transfom
The result can be stated in the form of the following theorem.
THEOREM 7. If f(^) L2 andc-p
F (a)g(Z) rmm(b) #fl F (a; b; -AZ) (detA)b-Pf (^)d^ (5.2)
^>0coverges absolutely in the simply connected region Re(a), Re(b) > p-I for Z > 0, and
then
Lim r (a) g( z) 0(z) (5.4)a+ m
The proof of the theorem is quite simple in view of the limit (see Herz [I]):
eim IFI (a; b; -R) rm(b) _p(R).a+
We shall now prove a theorem which serves as real inversion theorem or the gener-
alized Laplace transform.
THEOREM 8. If
r (a)m flg(a; b; Z)=Fm(b Fl(a; b;-Z)(det^)b-p f(^)d^
-A>0
is absolutely convergent for a > p-l, b > p-l, Z > 0, and f’ g uL2-p
(5.5)
thenbm
f( to llma H{g(a; b; ha) (5.6)r (a)
a m
where
(aAZ) (det Z) b-p g(a; b; Z)dZ.H{g(a; b; aZ)}-P
z>0
(5.7)
it’s Hankel transform will exist. So havePROOF. Since f e L-P
we
(Z) =f_p(AZ)(det^)b-Pf(A)d^^>0
(5.8)
Now, from Theorem 7, we have
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES 511
!z) (5.9)(Z)--lim "g(a; b;r (a) aa m
By the Hankel inversion of (5.8) and (5.9), we obtain
(a)f(^) lim F A_p(AZ)(det Z.B-p g(a; b; Z) (5.10)
a m
Now, changing variables from Z to aZ, and noting that the Jacobian of transformation
for the change is (a) pm we have the desired result.
REFERENCES
i. HERZ, C. S., Bessel functions of matrix argument, Annals of Mathematics, 61(1955),474-523.
2. JOSHI, J. M. C., Inversion and representation theorems for a generalized Laplacetransform, Pacific Journal of Mathematics, 14(1965) 977-985.
3. GARDING, L., The solution of Cauchy’s problem for two totally hyperbolic linear dif-ferential equations by means of Riesz integrals, Annals of Mathematics, 48(1947) 785-826.
4. ROONY, P.G., A generalization of complex inversion formula for the Laplace trans-formation, Proc. Amer. Math. Soc., 5(1954) 385-391.
5. YOSHIDA, K., Functional Analysis, Narosa Publishing HOuse, New Delhi (1979).