503
Internat. J. Math. & Math. Sci.
Vol. i0, No.3 (1987) 503-512
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.
rm<)
g(Z)
r m (8
flFl(
Our extension is given by
:8:- ^Z) f(^)d^
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
The function f(^) is
Let A be a symmetric matrix of dimension m
m
of a matrix.
orthogonal matrices.
where
0
E
0
m
the group of
m
m
is a symmetric function, then it is a function of the
f(^)
If
f(oAo’),
f(^)
called symmetric function if
elementary functions llke trace, determinant, etc., of A.
Herz [I] has defined the
Laplace transform with matrix variables by
(I.I)
etr(-^Z) f(^)d^,
g(Z)
^>0
where
^
E
Sm;
the space of
Z
X + iY, X,Y
X
(nljxlj); nlj
Lebesgue measure in
S*
m
the space of
m
being
Sm,
if i
err(A)
the set of all positive definite
f
(2i
m real symmetric matrices parameterlzed by
m
m
^
real symmetric matrices parameterlzed by
j, 1/2 otherwise.
stands for
e
trace
Also
^
Xo
is the
The inverse transform (I.I) is given by
err (AZ)g(Z)dZ
> 0
dA=i<_jH dlj
The integration in (I. I) is over
(A),A
0;
./
Re(Z)
(j),
0, otherwise
n
m(m + I)
2
(I .2)
R.M. JOSHI AND J.M.C. JOSHI
504
Herz [13 has used (I.I) and (1.2) in defining hypergeometric functions of matrix
The confluent hypergeometric function,
argument.
rm(b)
b; M)
IFl(a;
(2i)
rm(b)
Here
det(E- MZ) -a
fetr(z)
Re(Z)
X > 0
n
(det Z) -b dZ,
(1.3)
o
M
which holds for arbitrary complex
is defined by
IF1
and
Re(b) > m, provided we take
a
is the generalized gamma function of Siegel, and
X
Re(M)
>
(m + I)/2.
p
Herz [I] has shown that
r m (b)
b; Z)=
iFl(a;
IEetr(ZR)
rm(a)rm(b_a)
(det R)
det(E-R)b-a-PdR,
a-p
(1.4)
0
Re(a), Re(b) Re(b-a) > p-l.
In the present paper, we shall extend the generalized Laplace transform of Joshi
which holds for
[2],
to the space of
m
m
The kernel will be
real symmetric matrices.
IF1
function
We shall also prove some theorems on the properties of the transform
of matrix argument.
In section 3, we shall establish a relation between Laplace transform and
generalized Laplace transform through the operator of fractional integration defined by
thus defined.
3
arding
I
a
r-l,a)m
f(^)
I^f(R)
det(^-R)
a-p
(1.5)
dR,
0
Integral (1.5) is a generaliza-
f(R).
holding under suitable conditions on the function
tion of Riemann Liouville integral to matrix variables.
2.
DEFINITION.
Joshi [2] has defined the generalized Laplace transform in the scalar case by
r (a+b+c+l)(b+c+l)
g(x)
0
(xy)
b
IF
(2 I)
f (y) dy
-xy)
a+b+c+l
(b+c+l
In analogy with (I.I), we consider the relation
rm(a)
r-- f^
g(Z)
where
^
S*m
Sm, X, y
Z
converges in some
>0
and
(a; b;-^Z) f(^)
z
d^;
Z
x
is the function defined by
region, then it represents a function,
the complex symmetric matrix
Laplace transform of
IF1
F1
g(Z)
In this case,
there.
+ iy,
(2.2)
If integral (2.2)
(1.3).
g(Z), of the elements of
will be called generalized
f (^).
Now we have to establish the convergence of (2.2) to find the region where it defines
a function. But, before proving the convergence theorems, we have to say something on
the limit concept in matrix space.
We know that the space
defined in it by
A > B
$
m
<--> A-B is positive definite, where
A,B
a directed set since for any pair of matrices
that
A < kE
B < kE,
and
For any two functions
of the order of
g(^)
for
E
f(^)
being
and
g(A);
m
m
g(^)
infinitely large
and will write
f(^)
A
">"
is a partially ordered set for the order relatio
(),
S
m
A,
B
S
m
S
m
there exists a number
is also
k
such
unit matrix.
defined in
^
S
m
we will say that
f (A)
is
(in the sense of the order relation; >),
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES
f(^)/g(^)
if Moore-Smlth limit (Yoshida [5], p. 103) of
is equal to
3.
505
S
through the directed set
m
I.
CONVERGENCE THEOREMS.
Before proving the theorems on convergence of the integral (2.2), we will prove a
few lemmas which will be useful in proving convergence theorems.
LEMMA I(A)
IFI
(B)
If
Re(b-a) > p-l,
Re(b) > m, and
If
Fro(b)
(a; b; -M)
(a; b; M)~
-a
M
(=).
Re(a) > p-l,
Re(b) > m, and
IF1
(det M)
rm (b-a)
rm(b)
-(b-a)
etr(M) (det M
rm(a
(3.2)
(=).
M
We have, from (1.3),
1" (b)
m
f etr(Z) det(E- Mz
n Re(Z) =X
b; -M)
iFl(a;
(det Z)
-b
dZ
Re(b) > m
(2i)
rm (b)
(2i)
Now, if
-l)-a
(det M)
n
-a
/etr(Z) (det Z)
Re (Z)
X
-(b-a)
det(E-ZM- )-adz
(), applying formula (I.I) of Herz [I], we obtain
rm (b)
a
Re(b-a) > p-1.
1Fl(a; b; -M) (det M)
r (b-a)
M
m
This proves part (A) of the lemma.
To prove the part (B), we use
1Fl(b-a;
err(M)
b; M)
1Fl(a;
Kummer’s formula (Herz [1])
b; -M).
Kummer’s formula, part (B) of the lemma
Using part (A) of the lemma, and applying
is
proved.
LEMMA 2.
-AZ)(det
>b iFl(a;,b;
where
^)-P
rre(b) rm (6) rm (a-)
rm (a) rm(b-6)
dA
(3.3)
(det Z)
Re(a), Re(b), Re(6), Re(b-a), Re(b-6), Re(a-6) > p-l, and Re(Z) > 0.
From (1.4), we have
rm (a) rm (b-a)
rm (b)
where for
iFl(a;
b; -Z)
--fE
0
a-p det (E-R) b-a-p dR;
etr(-RAZ) (det R)
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 order of integration, we have
>0
wh e r e
8m (a-
iF1(a;
b-a)
This proves the lemma.
(a)
^)-P
d ^=
r m (b) rm(
rm (a)rm(b-a)
8m(a-,
b-a) (det Z)
is the generalized beta function of Siegel and
8m (a-,b-a)
THEOREM I.
b; -AZ)(det
rm(a-6) rm(b-a)
rm(b-)
Now we prove:
If the integral (2.2) converges absolutely for
Re(Z 0) < 0, then it converges absolutely for every
Z
for which
Z
Z
0
for which
Re(Z) > 0
provided
506
R.M. JOSHI AND J.M.C. JOSHI
Re(b) > m, Re(b-a) > p-l.
If the integral (2.2) converges absolutely for
(b)
Z
converges absolutely for every
Re(b) >
m.
(c)
a
Z
If
Re(Z) >
f
Now, as
Re(Z 0) < O,
where
Z
Z
0
then it
Re(a) > p-l,
then it converges for every
b;-Z)f(^)Id^--
f
b; -Z)
IFl(a;
iFl(a; b;-AZo)f(^)d^
F (a; b;-Z)
(3.4)
^0
^>0
^/()
from lemma I, we have
b; -AZ)
ilFl(a;
iFl(a;
AZ
b;
0)
Re(b) > m,
provided we have
l(det
Az)-al
(detAZo)-a
l(det
Re(b-a) > p-l,
Re(Z), Re(Z 0) > 0.
is bounded in the set of positive definite
vided
0
Re(Z0)
We have
llFl(a;
Z
Re(Z 0) < Re(Z) < 0, provided
for
b, and (2.2) converges absolutley for
for which
PROOF.
Z
Re(b) > m, Re(b-a) > p-l, and
ZO Z-l)al
But
iFl(a;
b; -^Z)
and does not vanish anywhere there, pro-
^
Re(Z) > 0.
Thus we see that
b; -AZ)
iFl(a;
iFl(a;
b; -AZ
0)
is bounded in the set of
Re(b) > m, Re(b-a) > p-l, and
p.d. ^, provided
Re(Z) > O.
Now (3.4) reduces to
f ll’Fl(a;
-AZ)E(^)Id^ <
b;
k.llFl(a;
-AZ0)f(A)Id^
b;
A>O
is the least upper bound (1.u.b.) of
^0
where
k
b; -AZ)
iFl(a;
iFl(a;
b; -AZ
0)
This proves part (a) of the theorem.
Re(Z) < O, Re(Z O) < 0,
Now, if
b; -AZ)
iFl(a;
iFl(a;
Re(b) > m,
Re(a) > p-l, we have from lemma l(b),
b-a
letr(-A(Z-Z0) )(det Zo Z-l) I;
b; -AZ 0)
So, as in the proof of part (a) of the theorem, we once again note that
iFl(a;
iFl(a;
b; --AZ)
b; -^Z
0)
is a bounded function in the partially ordered set,
Re(b-a) > p-l.
^
provided
Re(b) > m, Re(a),
So, it follows from inequality (1.4), that the integral (2.2) is abso-
lutely convergent.
This proves part (b) of the theorem.
We have yet to discuss the case
a
b.
In this case,
iFl(a;
b; -^Z)
is equal to
etr(-AZ), so that the transform (2.2) is reduced to the Laplace transform (I.I) which is
Z
absolutley convergent for all
is absolutely convergent for
THEOREM 2.
If
f(^)
Z
for which
Z O.
Re(Z) >
Re(Z0),
provided the integral (2.2)
This proves part (c) of the theorem.
is absolutely integrable in the partially ordered set
^
i.e.,
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES
507
sl^l
then the integral (2.2) is absolutely convergent for all
Re(b), Re(b-a)
>
PROOF.
iFl(a;
Since
Re(Z) > 0, Re(a),
Z, for which
p-l.
is bounded in the partially ordered set
b; -AZ)
provided
^
Re(Z) > 0, Re(a), Re(b), Re(b-a) > p-l, therefore we have
/IIFl(a;
b;-z)f(^)Id^_<
/IIFl(a;
A>O
A>O
A>O
in the partially ordered set
b; -AZ)
iFl(a;
is l.u.b, of
k
where
b;-^z)IIf(^)Id^_< k.IIf(^)Id^
A
> 0.
Re(Z)
for
This proves the theorem.
If(^)l
<
el (met)
a posltve number
and
d
If, for some number
THEOREM 3.
d-p
k,
;A > 0,
then the integral (2.2) is absolutely convergent for
Re(Z) > 0, provided
Re(a), Re(b),
Re(b-a), Re(a-d), Re(b-d) > p-l.
PROOF.
We have, from lemma 2,
IIIFI(a;
b;-AZ)f6
)Id^_<
/llFl(a;
k
)d-Pld^ <
^ >0
>0
Re(b),
Re(a),
where
b;-AZ)(det^
Re(a-d),
Re(b-a),
Re(b-d) > p-l, and
Re(Z)
> 0.
This proves
the theorem.
THEOREM 4.
If, for any
[f(^[
m
m real symmetric matrix
R,
[(det^)b-P[
< k. etr(-R^)
then the integral (2.2) is absolutely convergent for positive definite
R
provided
Re(Z+R) > 0, Re(b) > p-l.
PROOF.
/liFt(a;
b;
z)f(^)[d^
_<
I[iF1(a; b;-^z)[[f(^)[d^
<
^>0
^>0
<kfIiFl(a;
b; -^Z) etr (-R^)
(det^)b-P [d ^
<
,
A>O
from Herz ([i], formula (2.7)) provided we have
Re(Z+R) > O,
Re(R) > O,
Re(b) > p-l.
This proves the theorem.
4.
GARDING’S FRACTIONAL INTEGRAL AND GENERALIZED LAPLACE TRANSFORM.
In this section, we have derived certain connections between Laplace transform,
Hankel transform, and generalized Laplace tranform with the help of Garding’s fractional
integral.
From formula (1.4), we can show, by certain change of variables, that
r m (a)
r m(b)
where
1F
(a; b; -^z)(det Z)
b-p
rm(b-a)
etr(_RA)(detR)a-Pdet(Z_R]--,aR
fZ
o
(4.1)
Re(a), Re(b), Re(b-a) > p-l.
THEOREM 5.
If
g(a; b; Z)=
r m (a)
F m (’) f
IFI (a;
b;-^Z)f(^) dA
(4.2)
^>0
and
@(Z)
f etr(-AZ)(det
^>0
z)a-Pf(h)dA
(4.3)
508
R.M. JOSHI AND J.M.C. JOSHI
then
I
b-a
(Z)
z)D-P
(det
(4.4)
g(a; b; Z)
Provided integrals (4.1) and (4.2) converge in the generalized right half plane Re(Z)>
Re(b-a) > p-l.
and Re(a),
Applying the operator of fractional integration to both the sides of (4.3), we
PROOF.
have by changing the order of integration which is justified by Fubini’s theorem
I
b-a
fZ
F (b-a) ff ^)[ o
^>
q(z)
0(
m
b-a-p
dR] d^
etr(-R^)(det R) a-p det(Z-R)
Now, from (4.1) we have in view of (4.2),
I
b-a
(Z)
(det Z)
b-p
g(a; b; Z).
This proves the theorem.
From (4.2) and (4.3) we note that
a-p,
since
etr(-^Z).
a; -AZ)
iFl(a;
g(a; a; Z)(det Z)
(Z)
From Garding’s [3] theory, if derivatives of a function,
f(^) exist for suffi-
ciently high orders,
D
Since
a
i
a
f(^)
So from (4.4), we have
for all orders.
g(a; a; Z)(det Z) a-p
(Z)
D
for
S
b-a
where
^
Db-a[(det
(4.5)
[3]).
Moreover,
m
),
det(nij
if
being
D
i
j
and 1/2 otherwise
And for
z
S*m
det(3--ij).
z
Formula (4.5) has been established for real
which
Z) b-p g(a; b; Z)].
is the differential operator of fractional order (Garding
D
nij
f(^).
in equation (4.3) is a Laplace transform, the derivatives of it will exist
(Z)
Re(Z) > 0
by analytic continuation.
p-I
Z, it follows for all complex
Z
But we must note that in any case (b-a)
for
is
Re(a) > p-l. But the condition;
>
can be relaxed to the condition (b-a) > 0. To do so, we apply D-operator
(b-a)
p-I
to both the sides of (4.2) after multiplying it by (det z)b-P to get (4.5), but then
we must have; Re(a) > m, and b-a should be a non negative integer.
Formula (4.5) allows us to find a complex inversion theorem for the generalized
a non negative integer greater than
and
Laplace transform.
THEOREM 6.
If, for
rm (a)
F (b)
m
Re(a), Re(b) > m,
"[iFl(a;
b; -AZ)f(A)dA
g(a; b; Z)
jA > 0
is absolutely convergent in the generalized right half plane given be
met (X +iY) B-p g(a; b; X +
YSm’ X> X
o
iY)[dY
<
Re(Z) > X 0
and
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES
(X+iY)
det
YeS*
llm
X/((R))
b-p
g(a; b; X+iY)
509
0
dY
m
then, for A > 0,
(-l)m(b-a) Fm (b-a+p)
PROOF
(b-a)
g(a; b; Z)dZ
f(^).
0
m
provided
^z)b-P
b; ^z) (det
(2i) nF (b)
is a non negative integer.
(det Z) b-p g(a; b; Z)
(^), so that we have
Under the conditions of the theorem,
as a Laplace transform of a function, say
(det Z) b-p g(a; b; Z)
can be represented
etr(-^Z) (^) d^
(4.6)
0
Now, applying
D-operator to both the sides of (4.6), we have
Db-a[ (det z)b-P
>
g(a; b; Z)]
etr(-^Z)(det
Now, from Herz([l]), formula (I.I)), (det Z) -a-p
of
)a-2p
(det
rm (a-p)
provided
(^)d^
is the Laplace transform
Re(a) > 2p-I
property of Laplace transform, (det Z) -a-p D b-a
Laplace transform of
f(^)
-^)b-a
0
So, from the convolution
m.
(det Z) b-p g(a; b; Z)]
is the
la-P[det(-^) b-a
(4.7)
(^)]
Now, inverting (4.6) by complex Inversion formula of Laplace transform, (Herz [1]),
which is permissible under the conditions of the theorem, and then putting the values
of
@(^) in (4.7),
f(^)
we obtain
la-P[
(^Z) det (-^)
(2wilh fetr
X
Re(Z)
b-a
(det Z) b-p g(a; b; Z) dZ]
o
(-I)
re(b-a)
fla-P[etrZ)(det ^)b-a](det z)b-P g(a;
(2i)n
b; Z) dZ
Re(Z) --Xo
Now, from (4.1), we have
f(^)
(_l)m(b-a) rm(b-a+p)
n
rm (b)
(2i)
lFl(b-a+p;
Re(Z)
X
b-p
g(a; b; Z) dZ
b; ^Z)(det ^Z)
o
This proves the theorem.
COROLLARY.
If the conditions of Theorem 6 hold,
g(Z)
b-a
0
and
/etr(-^Z)f(^) d^
^>0
then
f (^)
r m (p)
F (b)(2i) n
m
b; AZ)(det^ Z) b-p g(Z)dZ
f
#IF1 (p;
Re(Z)
X
(4.8)
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
R.M. JOSHI AND J.M.C. JOSHI
510
variables.
5.
formula(4.8)reduces to Roony’s [4] formula.
m-- I,
In the scalar case;
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
(5.1)
f(R) dR
^>0
A
where
C
is the Bessel function of matrix argument
and
f
E
L2
C
L2
C
being the
Hilbert space of functions for which the norm defined by
- JR>
llfllc
2
where
c
If(R) 12
(dec R)
c
dR
0
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
L2
c-p
f(^)
and
F (a)
g(Z)
coverges
rmm(b) #fl (a;
^>0
F
b; -AZ)
(detA)b-Pf (^)d^
absolutely in the simply connected region
(5.2)
Re(a), Re(b) > p-I
for
Z > 0, and
then
Lim
a+
r m (a) g( z)
0(z)
(5.4)
The proof of the theorem is quite simple in view of the limit (see Herz [I]):
eim
a+
IFI (a;
b;
rm(b) _p(R).
-R)
We shall now prove a theorem which serves as real inversion theorem
or
the gener-
alized Laplace transform.
THEOREM 8.
If
rm (a)
g(a; b; Z)=
Fm(b
is absolutely convergent for
fl Fl(a;
b;-Z)(det^)
b-p
f(^)d^
(5.5)
-A> 0
a > p-l, b > p-l, Z > 0, and
f’ g
uL2-p
then
bm
f( to
llm
a
a
r m (a) H{g(a;
(5.6)
b; ha)
where
H{g(a; b; aZ)}
-P
(aAZ) (det Z) b-p g(a; b; Z)dZ.
(5.7)
z>0
PROOF.
Since
(Z)
f e L
-P
it’s Hankel transform will exist.
=f_p(AZ)(det^)b-Pf(A)d^
^>0
Now, from Theorem 7, we have
So we have
(5.8)
GENERALIZED LAPLACE TRANSFORM WITH MATRIX VARIABLES
(Z)--lim
a
511
!z)
(5.9)
"g(a; b; a
r m (a)
By the Hankel inversion of (5.8) and (5.9), we obtain
f(^)
lim
a
A_p(AZ)(det
F (a)
m
Now, changing variables from
for the change is
(a) pm
Z
Z.
B-p
g(a; b;
Z)
(5.10)
to aZ, and noting that the Jacobian of transformation
we have the desired result.
REFERENCES
i.
HERZ, C. S., Bessel functions of matrix argument, Annals of Mathematics, 61(1955),
2.
JOSHI, J. M. C., Inversion and representation theorems for a generalized Laplace
transform, Pacific Journal of Mathematics, 14(1965) 977-985.
3.
GARDING, L., The solution of Cauchy’s problem for two totally hyperbolic linear differential 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 transformation, Proc. Amer. Math. Soc., 5(1954) 385-391.
5.
YOSHIDA, K., Functional Analysis, Narosa Publishing HOuse, New Delhi (1979).
474-523.