ABSTRACT
Chapter 6
G-TRANSFORM AND MODIFIED G-TRANSFORMS
ON THE SPACE L
;r
6.1. G-Transform on the Space L
;r
This section deals with the G-transforms, namely, the integral transforms of the form of
(3.1.2):
Gf
(x) =
Z
G
m;n
p;q
"
xt
(a
i
)
1;p
(b
j
)
1;q
#
f(t)dt (6.1.1)
with the Meijer G-function G
m;n
p;q
"
z
(a
i
)
1;p
(b
j
)
1;q
#
dened in (2.9.1) as kernel. A formal Mellin
transformM, dened in (2.5.1), of (6.1.1) gives a similar relation to (3.1.5)
MGf
(s) = G
m;n
p;q
"
(a
i
)
1;p
(b
j
)
1;q
s
#
Mf
(1 s); (6.1.2)
where
G
m;n
p;q
"
(a
i
)
1;p
(b
j
)
1;q
s
#
= G
m;n
p;q
"
(a)
p
(b)
q
s
#
= G
m;n
p;q
"
a
; ; a
p
b
; ; b
q
s
#
=
m
Y
j=1
(b
j
+ s)
n
Y
i=1
(1 a
i
s)
p
Y
i=n+1
(a
i
+ s)
q
Y
j=m+1
(1 b
j
s)
: (6.1.3)
In this section on the basis of the results in Chapters 3 and 4 we charactrerize the map-
ping properties such as the existence, boundedness and representative properties of the
G-transform (6.1.1) on the spaces L
;r
and also give inversion formulas for this transform. As
indicated in Section 3.1, (6.1.1) is a particular case of the H-transform in (3.1.1) when
= =
p
=
= =
q
= 1: (6.1.4)
The numbers a
;; a
; a
; and given in (1.1.7), (1.1.8), (1.1.11), (1.1.12), (3.4.1) and
(3.4.2) for the H-function (1.1.1), are simplied for the Meijer G-function (6.1.2) and take
a
= 2(m+ n) p q; (6.1.5)
= q p; (6.1.6)
a
= m+ n p; (6.1.7)
a
= m+ n q; (6.1.8)
=
<
:
min
15j5m
[Re(b
j
)] if m > 0;
1 if m = 0;
(6.1.9)
and
=
<
:
1 max
15i5n
[Re(a
i
)] if n > 0;
1 if n = 0;
(6.1.10)
while in (1.1.10) remains the same:
=
q
X
j=1
b
j
p
X
i=1
a
i
+
p q
: (6.1.11)
Denition 3.4 on the exceptional set takes the following form.
