267
Internat. J. Math. & Math. Sci.
(1987) 267-286
Vol. i0 No.2
THE MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
E.L. KOH
Department of Mathematics and Statistics
University of Regina
Regina, Canada $4S 0A2
E.Y. DEEBA
Department of Applied Mathematical Sciences
University of Houston-Downtown
Houston, Texas 77002
and
M.A. ALl
#1598, Way 510
Muharraq 205, Bahrain
(Received June 2, 19 86)
ABSTRACT.
This paper extends the Meijer transformation,
(Mf) (p)
2p
(i+)
I
f(t) (pt)
/2K
(2 p)dt,
modified Bessel function of third kind of order
kernel,
e (-1, ) and K
,
is the
to certain general-
A testing space is constructed so as to contain the
(pt)/2K(2 p),
of the transformation.
function space and its dual are derived.
form,
given by
0
where f belongs to an appropriate function space,
ized functions.
M,
Some properties of the
The generalized Meijer trans-
f, is now defined on the dual space.
This transform is shown
to be analytic and an inversion theorem, in the distributional sense, is
established.
KEY WORDS AND PHRASES.
differential operator,
Meijer transform, generalized functions, Bessel
Schwartz distributions, Operational Calculus.
1990 MATHEMATICS SUBJECT
CLASSIFICATION CODES.
46F12, 44A15, 46F05, 33A40.
E.L. KOH, E.Y. DEEBA and M.A. ALl
268
In order to develop an operational calculus
INTRODUCTION.
i.
for the Bessel differential operator B
(-i,) and D
d’
t
Dt I+D,
where
Conlan and Koh [i] used the Meijer trans-
The operational formulas obtained
formation M f given above.
mirrored those obtained by Koh [2] by means of a Mikusinski-type
calculus.
The latter is algebraic in approach and allows for
certain convolution operators not covered by the classical
To remedy this situation, the Meijer
integral transform M f.
transformation has to be extended to generalized functions.
This is the object of the present work.
The idea is to construct a testing function space (
which contains the kernel
(pt)/2K(2 p).
The generalized
Meijer transformation M f is now defined on the dual space
q’
U,y
as follows:
(Mf) (p)
where e(-,l).
for feI’
2p
< f(t)
(pt)/2K
(2 p)>
(i.i)
The definition in (I.i) coincides
with the classical transformation whenever f is a regular
distribution, that is, one that can be represented by a suitable integrable function.
We note that there are various
extensions of classical transforms in which the operational
calculus is geared for other Bessel-type operators among which
are the K-transform [3] and the Hankel transform [3],
[4],
[5]
and [6].
In section two we shall describe the testing function
space
,
its dual and study some of their properties
While
in section three we will give the definition of the generalized
Meijer transform, show its analyticity in some region of the
complex plane, and then derive an inversion theorem.
Throughout the sequel we shall make use of the following
notations and facts and are stated for the sake of completeness.
MEIJER TRANSFORMATION
OF GENERALIZED FUNCTIONS
269
We shall denote the interval (0,) by I, the Bessel differk
ential operator
and its k-iterate by B
It
by B
tDtl-D
follows that
B
k 2k
[t D t +c tk-l_D t2k-i
I
k
_u(t)
k
+ctk-2.2k-2
D + +CkD t ](t),
where the c’s are suitable functions of
The modified Bessel functions I
(i 2)
u only.
(2 p) and K
(2 p) of first
and third kinds are defined by (see Watson [7])
I (2 p/)
(pt) k+/2
k’F(k+l+)
k=0
[
any real number,
(1,3)
and
si Z
(pt) k-/
k!F
(k+l-)
k=0
pt) k+/
k=0
K (2 p)
(-l)k(-b-k-1)
1
[-logc (-)
c
(pt) k+/2 +(-i)
k’
k 0
+1
not an integer,
k! + i+)
1 + -+k i
[ T)
i=l
i=l
-
(pt)k-/2
k=0
k’
(-$-)
integer
(1.4)
( is Euler’s constant) and the finite sums are taken to
e
be equal to zero whenever the upper limit is less than the lower
limit.
Further,
B
B
k_ ((pt) /2K (2 p))=
k
p (pt)
k
k
p (pt) /2 I (2 p)
((pt)
/2 I (2 p))
/2K
(2 p/),
(1.5)
(i 6)
Finally, the following asymptotic expansions of K
and I
-
(2 p) will be used repeatedly (see Watson [7])
K (2 /)
and
(2-[)
"-
(pt)
e-
i+0
pt -1/2 ],
-
<
argp
.
(i.7)
E.L. KOH, E.Y. DEEBA and M.A. ALI
270
l(pt)-i/4(e2 P+ie-2 P+i)
2/{
[l+0(Iptl -I/2) ],
3n
-/2 < argp <
T’
(1.8)
-1/4(e2 p_i e -2 p-i
[l+0(Iptl -1/2
-3/2 < argp < /2.
The Testing Function Space
2.
M
In this
and Its Dual.
section we shall define the testing functions space M
,y’ its
dual and study some of their properties.
be any real number and e(-,l).
Let
,,y
{eC
Define
(I)lly,k(#)<}
where
sup
l’k (#)
0<t<
le
In view of the weight function
eY/tl-one
may think of the
as those smooth functions on I that grow no
elements of
faster than the exponential function e Yet for large t and
behave like power functions near the origin.
As will be seen
in Lemma 2.1 below, the kernel of the Meijer transform is of
this type.
The family {I
y,k
is a countable multinorm
k:0 1 2
Indeed, 1 y,k is a semi-norm for each k and 1 y,0 is a norm on
We assign to
the topology generated by this family
,Y
,Y
of multinorms.
Thus
is a countably normed space (a fund-
amental space and members of the dual space M’
functions).
all
9
A sequence
and for every k
independently.
{#9}
are generalized
e
is Cauchy in
for
0, I, 2,
y,k
We shall now show that the kernel,
(pt)/2K
(2 p),
member of
Lemma 2.1.
For any fixed complex number p such that p
arg p < n and
Proof.
Re2/
> y,
(pt) /2 K (2 p)
e
0,
,
Under the hypothesis on p, K (2 p) is an analytic
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
271
function of t on the right half plane and hence a smooth function
It remains to show that l
on I.
1
From (1.5)
0, i, 2
k
y,k
((pt)/2K
y,k
((pt)
/
’2K
(2/))
we have
suple/tl-pk(pt)/2K
(2 p))
(2 p)
tEI
We will consider two cases:
t small and
for
<
I.
(2 I)
for small t and for large t.
For
not equal to an integer t/e asymptotic expansion
of K (2 p) in (i 4) yields
-<AReY/t IPl k+-i [IPtl i-+ iptl 2-+
eYtl-pk (pt) W/2K (2 p/)
CeY/Iplk+-l[Iptl+Iptl2+Ipt 13+
+
where A
and C
are functions of
hand side of (2)
for
(2 2)
Since W < i, there
only.
exists a real number T < 1 such that for
ptl
is bounded by a constant.
0, -i, -2,
le/tl-pk(pt)/2K
< T, the right
For t small for
the expansion in (1.4) yields
_<A,eY/
(2 p/)
k+-i
2
+
CeY/{Iplk+-lllogC(pt)
[Iptll-+ltl2-ptl3-+...
+
Ee/l
pt
where, A’
p k+-i
C’ and E
+
pt
i-+
2-+
pt
3-+...
are constants depending upon
(2.3)
only.
Applying L’Hdpital’s rule to the second term in (2.3) and
using the fact that
< i, we imply that there is a real
number T < 1 such that for
Iptl
< T, the right hand side of
(2.3) is again bounded by a constant.
is bounded for small t and
Thus the left side of (2.2)
< 1 for all k =0, i, 2,
-
We now consider the finiteness of (2.1) for large t.
employing (1.7), we obtain, for
le/tl-p k(pt)/21<
(2 p)
[I+0
< E’
< arg p <
and for
Iptl
By
> T,
Iplk+-le(-Re2/)/{Iptl3/4-/2
IPtl-i/2)]
E.L. KOH, E.Y. DEEBA and M.A. ALl
272
where E’ is a constant depending upon
2,
and y< Re
only.
< i
the right hand side of the last expression is
Iptl
bounded by a constant for
> T.
Thus in either case we have shown that
0, I, 2
for all k
Since
y,k
(pt)/2K
Therefore,
((pt)/2K
(2 p)) <
(2 p)eM ,y"
By employing the following fact
Dt[(Pt)+/2K
-p(pt)-+/2-1/2K
(2 p)
-
(2 p)
and by arguments similar to the proof of Lemma 2.1, we have
For p
Lemma 2.2.
Dt[(Pt) /2K
k
that
0,
(2 p)] e M
Dpk[(pt)/2K
< arg p <
and Re
2/
> y,
Further, by symmetry, it follows
(2 p)] e
(D
,Y
k
is the k-iterate of D).
In the next few lemmas we shall investigate some properties
of the space M
Lemma 2.3.
Proof.
y,k (#)
subset
<
and its dual.
,Y
The space
Let {#
9
is complete.
be a Cauchy sequence in
for all k
K of I,
,Y
k
B_()
Then
,y
Hence on every compact
0, i, 2
converges uniformly
for each
Let (t) be a smooth function on I and s
rt
a fixed point of I. Define
| (T)dT. Then
s
D
(s). It can be verified that
(t)
(t)
0, i, 2,
k
(D-I)(t)
-1Dt
t
l+D-lt-B_(t)
s
()
Dt(t)
l-Ds (s)
(2.4)
and
D-It-I+D-It-B _(t)
(s)
(t)
L s(t)Ds$ (s)
(2.5)
where
L s(t)
Fs -(t-s
0
s(int ins)
0.
Since multiplication by a power of t or the application of D -i
preserves uniform convergence on K, it follows from (2.5) that
Ds$ converges
on K.
From (2.4), it follows that
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
Dt
Dt v
D
v
by B
-- -
converges uniformly
Repeating the same argument with
and B
k+l
and B
respectively, we deduce that
#
k
0, i, 2,
k
Now the convergence of
on K as +=.
converges uniformly
Dt#
Dtk(t)
Since
as +.
{#}
on I such that for
Dtk(t)
converges to
< e/2.
(-)+0
Further, since
eM
it follows that
+.
as
e
> N
Since 1
Hence
,Y
,
such that if
k
we have for each teI and
y,k
y,k
,Y
>
,
is clear that
is a subspace of
restriction of fe’
7’r6tl-Bk_(-)l
that
k
that there is a real number
is in
’
,Y
if y >
The real number
0 as
leYCrtl-B
() < 1 y,k
k
( -#)
) + 1
(
y,k
()
is a complete space.
M ,Y
, belongs
to
,Y
,y
(t)M,y.
Nk,
(Indeed M ,y is a Fr4chet space)
Now if y and e are any real numbers with y <
to
/.
We will be done if we can show that
> 0 there is N
Iy,
k (-)
,
for each
is a Cauchy sequence, it follows that for each k
and each
Thus 1
and each tgI,
0, i, 2,
replaced
#
on K as
Therefore there exists a smooth function
each k
-
imply the uniform convergence on each K of
and B
k
273
to
,
then it
Further, the
’,"
This implies
such that the restriction of f
f
and is not in
if y <
f
,y
f"
’
is called the abscissa of definition
of
of f.
For any real numbers y and e with
M,e
is a subspace of
Lemma 2.4.
of
,
However,
,Y
If y <
,
< e, we know that
is not a dense subspace
then
M ,Y
Proof.
Let (t) be a smooth function given by
(t)
We claim that (t)e
%"y,k (#)
<
sup
l<t<2
,Y
I
0
e-7/t-l+u
Indeed,
le/tl-UB k-la (t)
+
sup
2<t<
-
IeY/tl-UB k
#(t)
(2 6)
E.L. KOH, E.Y. DEEBA and M.A. ALl
274
is a smooth function, it follows that the first
Since (t)
Moreover,
is bounded.
term on the right hand side of (2.6)
from (1.2) and the observation that
D
t(e- /t-l+
k
Pk(t
where
[Pk(t
-i
and
+qk-i (t -1) t
-i
qk-l(t
-1
-l+e-/
-I/2 t
are polynomials in t
-i
Thus, the second term on the right hand side of (2.6)
Hence
,k
given by
N
N
(#) <
and
e
M ,y
{eM1,yl.
O 1/2 ()
is bounded.
Let N a neighborhood of # be
(-) <i/2}.
y,0
X,O (-) < 1/2 implies
ee/tl-@(t) e e/tl- l(t) >
NOW for large t,
that
Thus
(e-Y) /
l(t)
>
21--e-Y/t -I+
which implies
Hence no element of
,o () remains unbounded as t+
N belongs to M
Therefore
is not dense in M
,y"
that
,
,
Finally, we shall discuss the adjoint of the operator
B
k
It is rather clear that B k
operator on M
,Y
*k
k
adjoint B
of B
<B *k f
-
Let feM’
is a continuous linear
e ,Y
and
,Y
We define the
by the equation
>
<f
Bk
>
k
0, l, 2,
-
The continuity and linearity of B
will imply that B *k fe’
,y
*k is
and B
a continuous and a linear operator on
Further
,y
*k
in lieu of equation (1.2), the expression for B
is given by
*k
B_
’
k
CkD t
+
k+l +
Ck_itDt
-
where the C’s depend only on
+
.
B*
B
tD
+
k
t D 2k
t
Indeed if k
+ (I-)D t while B
B
1
then (i 2)
+ (l+)Dt)
tD
Inductively, we obtain B *k
k
We end this section by the following remarks.
Remark 2.1.
f(t)e-Y/t -I+
If f(t) is locally integrable on I and
is absolutely integrable on I, then f(t)
generates a regular member f of
(2.7)
Moreover, we can show that
the adjoint operator B *k is B k
yields B
.k-I 2k-i
Dt
Clt
’
,Y
via
Thus
275
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
f (t)
<f,>
#eM ’Y
(t)dt,
0
(2.8)
The space D(1), of all smooth complex valued
Remark 2.2.
functions on I whose support is contained in a compact subset
K of I equipped with the seminorms
(t)
suplD
teI
Pn ()
K is a subspace of M
,y
Further, the restriction of any fe
to D(I) belongs to
where #eD(I) is such that supp
,7
D’ (I).
3.
The Generalized Meijer Transform.
In this section we shall
give the extension of the Meijer transformM f to generalized
functions belonging to
’
We shall also prove the analyticity
,Y
of the transform and exhibit an inversion theorem which is a
generalization of the classical inversion formula (see [i])
For fe’
e(-,l) and
,Y
pelf
{PEIRe 2/
> y >
o,r
p
arg P l<n}, we define the Meijer transform of generalized
function by
2P
(Mf) (p)
F’(I-)
< f(t)
(pt)/2K(2 p)>
(3.1)
In (3.1) if f(t) is a regular generalized function (see Remark
2.1) then we obtain the classical Meijer transform
(Mf) (p)
r (I-)
I
f(t)
0
(pt)/2K(2 p)dt
where Je(-l,).
Theorem 3.1.
Let
transform in (3.1) for
f&.(’,y and (%f)
pef.
Then
k
S (B f)(p)
Proof.
For
(Bkf) (p)
pelf,
2p
k
pk (f) (p)
0, i, 2,
<
(p) be the Meijer
Bkf(t)
we have
(pt)/2K
(2 p) >
(3.2)
0
E.L. KOH, E.Y. DEEBA and M.A. ALl
276
-
k
(pt)/2K
r (i-)
2p
< f(t)
B
2p
< f(t)
pk (pt) /2 K (2 p)
F’(I-)
pkM
(2 p) >
>
(f)(p)
by equations (1.5) and (3.1) respectively.
Theorem 3.1 is the basis for an operational
Remark 3.1.
calculus for the operator B
It is the generalized function
analogue of the differentiation theorem of Conlan and Koh ([i]
Theorem 2, page 149).
We shall next prove the analyticity of the transform
However, we need the follow lemma.
M (f)
Lemma 3.1.
point in
f.
Then
eY/ (pt) I- /2K
sup
0<t<
Proof.
Lemma 2.1.
,
as t
(2 p)l
<_
c (+Ipl-/2+ 4
The idea of the proof is similar to that of
Since K (2 p/) is analytic except at t
we need only to show the boundedness of
0 and as t
K(2 p/)__
m.
(1.4) with
leY/E(pt) I-/2K
such that
0, -i, -2,
(2 p)l < A e
YF
AI Y[Iptl+Iptl=+Iptl +..-]
only.
For T < 1
< T, the right hand side of the last expression
is bounded by a constant
if we employ (1.4) with
R
independent of p and t.
ey/E(pt) i-/2 K(2 p)
is independent of p and t.
Similarly,
then, for Iptl <
0, -I, -2,
we have
where Q
eY/(pt)l-/2K
we obtain
and A’ are constants depending upon
Ptl
0 and
Employing the series expansion of
+
where A
(3.3)
).
.
is a constant depending only on
where C
t
Let y be any real number and p be a fixed
<
Q
T,
(2/)
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
277
For t large we employ (1.7) to obtain
le/(pt) I-/2K (2 p)
where L
3/4e (y-Re2)
/2+
L1l pt
_<
is again a constant independent of p and t.
E max{R,Q,L}.
Then
(pt)l-/2K
sup
(2 p/)
< E
0<t<
Re2/
and since
U
(l+Iptl-/2+ 3/4 e (y-Re2)/
EU (l+IPl -u/2+3/4) (l+Itl
<
/2+
3/4 e (v,-Re2)/
> y and pe(-,l), it follows that the right
hand side of the above expression is less than
with C
P
a constant depending upon p.
U
Let feW’
Theorem 3.2.
pef.
Dp(Sf) (p)
Proof.
(pf)(p)
Then
For
2
F(1-b)-
is analytic in
< f(t)
Dpp(pt)
/
Namely,
the Meijer transform
f
and
2Kp (2 p/{)>.
C be the circle about p of radius r so
pelf,let
chosen that i lies entirely in
some r
(f) (p) is
and
Cp(l+Ipl -p/2+3/4)
This completes the proof.
We will now show that M f is analytic.
of f and
Let
f
and let
gpl
< r
I
< r for
I
(f) (p+Ap)- (f) (p)
2
Ap
<f(t),D
P
p(pt)/2K U (2 p)>
<f(t),#Ap(t)>,
where
(t)
2
(i-p)
(p+Ap)[(p+Ap)t] 2K
(3.4)
(2/(’+AP-p(pt)2K (2/)
Ap
Dpp(pt)
2Ku (2)
].
show that
In order to prove the result, it suffices to
as Ap
0 in
U,y.
it follows that (3.4)
k=0, i, 2
Since by Lemma 2.2
is well-defined.
Ap(t)
0
Dpp(pt)O/2K(2/)e,,
Further, since for
278
B
k
E.L. KOH, E.Y. DEEBA
Dpp(pt)/2K v (2 p)
by equation (1.5)
of p on
f.
D
p
and M.A. ALl
pk+l(pt)/2K v (2)
it follows that B
k
is an analytic function
-U Ap
By Cauchy Integral formula, we have
Bk
(t)
-V Ap
Ik+l
1
iF (i-)
(t)
V/2K
(2 /f)
C
1
1
(_p_ A--Ap
iF (l-b)
Since
I-Pl
l-p-Apl
r
leY/tl-B
2]dE
k+l(t) V/2K(2)
C
(-P) (-p-Ap)
2
> r- r
I -/2 3/4
and
I
_
eY/(t) I-V/2K
(2 /6) is
it follows that
CI.Apl
k
(-p)
,:
+
bounded by C (l+
1
_p
f
(l+ll
2
+34
C
2
Ap,
F(I-) r(r-r I) sup
<
c
p
which converges to zero as
(II k+ +II k+/2+3/4
0 and this completes the proof.
The rest of this section is to prove a generalization of
the inversion theorem for the classical Meijer transform (3.2)
due to Conlan and Koh [1].
Our result is restricted to generalized
functions belonging to the space D(I)
(Remark 2.2).
In particular,
we shall prove the following inversion theorem
Theorem 3.3.
fe’
for
f(t)
lim
V,Y
Ol
pefif.
F(I-)
2i
2
where p(O)
(Mf)
Let
(p) be the Meijer transform of
Then
I%
-O
1
20
iO sec
(vf) (p)p-l(pt) -V/2I v (2p) dp
and
8
is a fixed real number in
f.
(The limit is to be understood in the sense of convergence in
D’ (I)).
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
279
The proof of Theorem 3.3 will be developed in a sequence
of lemmas.
We will state the lemmas and give them their proofs
in the Appendix.
Lemma 3.2.
Let eD(I) and feM’
Then, for a fixed,
,Y
81 (0,),
1 A(p)p < f(t)
1
<f(t),
-%
(pt)/2K(2 p)
I
8
1
> dp(8)
A(p)p(pt)/2K
(2 p)dp(8)>
1
where
I
A(p)
#(t)
(pt)- /2p- 1 I (2) dt,
0
y >
max(0,of)
Re2
and
Let ED(I) be such that
Lemma 3.3
be a real number such that 0 <
such that
> 0.
Re2/
y,
Q2,81 ()
< A.
supp
[A,B] and
Then for any Y > 0
set
I
/-{+( 2
(/_) 2
L
81 (t,)B_k(t)dt,
where
L 8 (t )
1
[ t-/2u/2
(/-Iu+I (2 p)K
t-r
+ /I (2 p)
Then
e/
WSl()
<
to zero on 0 <
on 0 <
<
e
Y/
as @
Lemma 3.5.
p (-81
converges uniformly
1
QI,8 I()
S%1()
p(81)
Under the same hypotheses of Lemma 3.3 let
Lemma 3.4.
Then
K+I (2 p))]
I-[Q2’81()-Bk-#()]
.
as @
(2 p/)
I-QI,
.
I
8
0
LSl(t’)Bk-(t)dt
() converges uniformly to zero
1
1
Under the same hypothese of Lemma 3.3, let
E.L. KOH, E.Y. DEEBA and M.A. ALl
280
I
Q3,8 1 ()
Then R 8
L 8 (t,)B
1
-
k
(t)dt.
converges uniformly to zero
.
<
on 0 <
(/+
e7/ I-Q 3 ()
’81
as 81
()
i
2
Finally,
H81
Lemma 3.6.
Let #(t) eD(I).
e1
K (2 p/)
I -81 (p)/2
()
HI()
converges in
MU,
181
2hi
0l
"
Let (t)eD(I).
(f) (p)p-l(pt)-/2 I
-8
I (2 p)dtdp(O).
#(t) Cpt)
0
to () as
Proof of Theorem 3.3.
<F(1-)
i
let
real number y
positive
Then for any
Re2/,
For a fix
We show that
(2 p)dp(8), (t)>
(3.5)
1
converges to <f(t), #(t)> as
I (2 p-E)is
Since (pt)
81
a smooth function on I, the integral in (3.5) is locally
integrable with respect to t, it follows that (3.5) can be
written as
F(I-)
2i
I
1
-e 1 (f)
#(t)
0
F(I-)
2i
; 81 ;
-e 1
(t)p
(p)p-l(pt)-/2i(2 p)dp(e)dt
-i (pt) -/2
0
I (2 p) "F(I-)
dtdp
by Fubini’s Theorem and (3.1).
that
<f(),
>
max(0,f)
<f()
(3.6)
e
For any real number 7 such
to
Lemma (3.2) implies that (3.6) is equal
/2K p)J0 , (t) (pt) -/2 I
r
el
(p)
1
(p)/2Xp/)>_
(2
(2
,
M,y
(T) > in
<f (T)
which converges uniformly to
imply that
Lena 3.6. Equation (3.5) will then
(2 p)dtdp(8)>
as
el
n by
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
f(t)
F(I-u)
lim
"-2i
%1/
rJSl_81
(
2i
)(P)P-I(pt)-U/2Iu (2 p/)dp(8)
which completes the proof of the theorem.
Remark 3.1.
(Muf)(p)
If
transforms of f and g for
(u f) (p)
(Mug)(p)
for
equality in D’ (I).
(Mug)(p)
and
pelf
peg
respectively and if
then f
g in the sense of
and
pefng,
are the Meijer
This uniqueness property is a consequence
of Theorem 3.3.
APPENDIX.
In this appendix we shall give the proofs of
Lemma 3.2 through Lemma 3.6.
Proof of Lemma 3.2.
A(p)p(p)U/2K U (2 p/)dp(8)
N(Y)
_%
Set
1
Clearly N()EM
U
XY’k (N (r))
<
This follows from Lemma 3.1
1
C
-8
.
IA(p) Ipl k+u(l+Ipl -]/2+3/4) Idp(8)
< =0.
(A1)
1
Let R(t,m) be the Riemann sums for N() given by
m-i
A(p(
))p(
(Yp(
)/2K
(2/
3=-m
We need to show that for feM’
,Y
,<f(t) ,R(,m)> converges uniformly
to <f (t) ,N () >.
Equivalently, since fe;’
,’ we need to show
that R(,m) converges in M
to N() as m
That is,
U,Y
Y/?
e
G(,m)
TI-Bk
(N(Y)-R(,m))
converges uniformly and absolutely to zero on 0 < r <
Since l U
(N(T)) < it follows that for a given
r,k
> T
choose a real number T such that for
suple Y/
>T
and for all m
-
I-UBk
N()
< e/2
as m
> 0 one can
(A2)
.
282
suple
T
-
l-Bk
>T
E.L. KOH, E.Y. DEEBA and M.A. ALI
< e/2(
R(T,m)
18
1
iA(p
idp(8))-i
-8
8
m
m-i
(A3)
which implies that there is m
such that for all m > m
(A3) is
0
0
Thus from (A2) and (A3), we have sup G(,m) < ,for
less than e/2.
T>T
K
(2-) as
e/pk(pT)l-/2K
0
T
to zero on p(-8
I)
< p(8) <
p(81).
(2 p) converges uniformly
If we redefine this function
Y/ k
p (p)
to be zero at
0 for the given p(8), then e
becomes uniformly continuous on (T,p) with 0 <
p(-8 I) < p(8) <
p(81).
I-/2K
(2 p/)
< T and
Thus R(T,m) will converge to the
integral and hence there exists m
>
I such that for all m
ml,
sup (G(T,m)) < e.
0<<T
Therefore, for all m > max(m 0,m I)
sup
O<T<oo
That is, G(,m) converges uniformly to zero on 0 <
<
as
In the next three lemmas we shall investigate the boundedness
of the integral
IL
0
where L
8
8
(t,)B
1
-
k
(t)dt
(t,T) is as defined in Lemma 3.3.
1
of Lemmas 3.3, 3.4 and
3.5
Since the proofs
are essentially similar, we shall
only give the detailed proof of Lemma 3.3.
The idea of the
proofs is to use the asymptotic expansions of K (2 p) and
I(2 p)
that
and get appropriate bounds
P (81)
on the resulting terms.
Proof of Lemma 3.3.
Since supp (t)
Q2,81 (T)
(/?+c)2
(r+s)2.
we consider
2
m>.
< T and by employing the series expansion of
Now for 0 <
0 if either
(/_c)2
< T <
81
e i81 sec 2 --,
< A or
and
12 pl _> I’
sec
el
-I
(/_c)2
Further, since
it follows that
81
[A,B], it follows
> B.
Thus
MEIJER RANSFURMATION OF GENERALIZED
(see (1.8))
283
(2p- (see(l.?)
Employin the asymptotic expansions of K
of I (2 p)
FUNCTIONS
and
we have
W8 1 ()
Jl
+
J2
+
J3
+
(A4)
J4’
where
3__U
1
Jl
4
t
2
2
4
e
sin (
81
(/-/) tan-)
2
B
k
(t)dt
y3/4-/2
I
t-/2-1/4eY (/-/) siny
(’?+’) 2
1
B
k
(t)(0(Ipt
1
i
81
(/{-/T) tan-)
1
2)+0(Ip 2)+0(Iptl) 20(Ip
2)at
and
y ,/’"
J3
3/4-/2 [l+0(Ipl -1/2
(/T+ 2
B
J4
_e
k
(t)
81
t-/2-i/4 e -7(/+/) cos ( (/{/ tan[l+0(Iptl-1/2)]dt
-
YF l-Bk
-
()
We show now that (A4) converges uniformly to zero
on 0 <
.
<
Adding
Jl
and
J4’
F- /T
replacing x
and adding
and subtracting the term
-el
y/T
l-Bk
#()
81
I
sin(yxtan-)dXx
we obtain
i
I
+ e Y/{
_
-
v (x, t) sin (yxtan--) dx
l-Bk
#()
1
[
I
e1
sin(yxtan--) dx
x
i]
(A5)
28
E.L. KOH, E.Y. DEEBA and M.A. ALI
where
eYX(X__
v(x,)
e
Y/
+
I)-+I/2B k ((x+/{)2)_B k
()
/{
iT
x
-
v(x,T) has a removable singularity at x
0 and is bounded
/
/ 2
/
A
< r < (/ +
on the domain
< x <
(x,Y)I--) }. So
,
given any e > 0 we can make the first term in (A5) less than
e/3 by choosing
small enough, say o
of (AS) converges to zero as
In
J2
.
81
the terms represented by
are bounded by
C/ll+i
tan
81
I
The second term
i"
J1 + J4 < e/3.
0(Iptl -I/2) and 0(IpI -1/2
Thus
for C sufficiently large.
This
implies that
sin (y
and
sin (y
81
(/-/) tan)
01
(/-/) tan--)
0({ptl -I/2)
0(I
< C’
(P)-I/21)
< C"
where C’ and C" are constants independent of 8
I.
Therefore the
This implies that given any e > 0, we can choose
sufficiently
integrand of
is bounded on the domain
J2
(t,)
small, say o
IA<t<B; (/_)2<< (/+)2}.
2
such that
IJ21
<
e/3.
By a similar argument, for e > 0 we can choose
sufficiently small, say
Let
3.
min(l,O2,o3).
J31
such that
e/3.
Then for a given e > 0 there exists
a 6 > 0 such that w (T) <
whenever
8
1
w A (r) converges to zero on 0 < T <
as
1
Finally, we shall prove lemma 3.6.
181-I
Proof of Lemma 3.6.
<
< 6.
.
81
Equivalently,
Since (t) has a compact support and
since the integrand is smooth, it follows that
-
Bk
H8 1 ()
1 Bk
-%1
(pr)
(2 p)
(t)
(pt)-/2I
(2 p/)dtdp(8)
285
MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS
Using (1.5)
B
(1.6)
and
we obtain
k_UHSI() -! I@l-81(P) U/2K
]0f=(t) (pt)
(2 p)
-U
k
B
(pt)U/2I
(2 p)dtdp(@
Integrating the inner integral by parts 2k times, we get
B
k
H
1
()
(p)
(Pt)-/2I
(2 p)
(2)B k (t)dtdp(0)
_
and upon interchanging the order of integration
-
Bk H
81
(%)
(pT)
0
-8
K (2
p)(pt)-/2I
(2 p)B k (t)dp(8)dt
1
Now, by observing that
(p)/2(pt)-D/ 2} (2 p)I (2 p/)dp(8)
(t,T)
(see Erd41yi [3], Equation 9 page 90) where L
8 (t,) is as in
1
Lemma (3.3) we obtain
B
k_H 8 1 (T)
0
Ls_(t,T)Bk_
1
-
(t)dt.
If supp(t) is contained in [A,B), then, for o a real number
such that 0 < o < A
BkH8
I
(%)=
Bk H
8
(/{-o) 2
QI,8 I()
where
Qi
8
()
i
l
respectively
(T) can be written as
1
Now e
dt +
+
;
(/+o) 2
dt +
;
Q2,01(%) + Q3,81(%)
i, 2, 3 are as in Lemmas 3.4
"f/’
uniformly to zero on 0 <
Semma 3 5 respectively
dt
I-QI ,81()
<
as
Moreover
and e
81
e
/
Y/?
3 3 and 3 5
i-
Q3,81
converges
by Lemma 3.4 and
i%
()_B k
[Q2,Ol
_(%)]
E.L. KOH, E.Y. DEEBA and M.A. ALl
286
converges uniformly to zero on 0 <
HSI
() converges in
.{,y
to () as
<
01
as @
.
z.
Hence
1
This completes
the proof of Lemma 3.6.
ACKNOWLEDGEMENT.
The work of the first author was partially supported
by NSERC of Canada under Grant A-7184;
a grant from the University of
that of the second author by
Houston-Downtown.
REFERENCES
[I]
Koh, E.L.,
A Mikusinski calculus for the Bessel operator B
Lect. Notes #564(1976), 291-300.
E__q., Springer-Verlag
Proc. Diff.
[2]
Conlan, J. and E.L. Koh, On the Meijer transformation,
and Math. Sci., Vol.
(1978), 145-159.
[3]
Zemanian, A.H.,
New York, 1968.
[4]
Dube, L.S. and J.N. Pandey, On the Hankel transform of
distributions,
Tohoku Math. J., 2__7 (1975), 337-354.
McBride, A.C., The Hankel transform of some classes of generalized
functions and connections with fractional integration. Proc. Roy.
Soc. Edinburgh, 81A (1978), 95-117.
Koh, E.L., The n-dimensional distributional Hankel transformaeion,
423-433.
Can. J. Math., 27(2) (1975)
[5]
[6]
[7]
[8]
Internat. J. Math.
Generalized Integral Transformations, Inter-Science
Watson, G.N., A Treatise on the Theory of Bessel Functions,
University Press, Cambridge 1966.
Erdlyi, A, et al, H.igh. er
..Tr.an. sce.n.dental .F.u.nce.io. ns,
McGraw Hill, New York, 195,3.
Vol. II,