Internat. J. Math. & Math. Scl.
Vol i0, No.3 (1987) 417-432
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
RICHARD D. CARMICHAEL
Department of Mathematics and Computer Science
Wake Forest University
Winston-Salem, North Carolina 27109
U.S.A.
and
R.S. PATHAK
Department of Mathematics
Banaras Hindu University
Varanasi 221005
India
(Received April 4, 1986 and in revised form October 29, 1986)
Initial and final value Abellan theorems for the Whittaker transform of
functions and of distributions are obtained. The Abelian theorems are obtained as
in absolute value inside a
the complex variable of the transform approaches 0 or
wedge region in the right half plane.
ABSTRACT.
KEY WORDS AND PHRASES.
Distributions.
Abelian Theorzm, Whit2.akzr
1980 AMS SUBJECT CLASSIFICATION CODE.
Transform of
Functions and
44A20, 46F12.
INTRODUCTION.
1.
Abelian theorems for the Whittaker transform of functions and of distributions
are obtained as the complex variable of the transform approaches 0 or
value inside a wedge region in the right half plane.
The setting for these Abelian
theorems is motivated by the initial and final value results of Doetsch
3h]
33 and
zation.
[2,
[6]
in absolute
[1,
sections
for the Laplace transform of which the Whittaker transform is a generali-
Our results here generalize Abelian theorems previously obtained by Zemanian
8.6 and 8.7], Akhaury [3-4], Moharir and Saxena [5],
sections
and Tiwari and Ko
in the generality of the transform variable being in a wedge and in the generality
of the parameters.
THE WHITTAK2_2 FUNCTION.
2.
Let k
(ml-kl+(i/2))
For s
>
Wk, m( s
PK
>.
and m
kl+ik2
> 0.
ml+im2
Let s be a complex variable with s e
the Whittaker function
e -s/ s
be complex parameters which satisfy
m+(/)
rlm-ki(1/2-))
0
e
Wk, m(s)
-su
m-k-(1/2
u
Let p be a positive real parameter.
<
{s
s > 0 and
e
Sl+iS 2
Using
(2.1)
’
I
with the gamma
(Erdlyi et
al
{s
#>
sl+is 2
e
[7, (2), p. 264])
(l+u )re+k-(1/2
:
s
I
> 0}.
is given by
du
(2.1)
For K > 0 being a fixed real number put
in the right half plane
Is21 KSl); PK is a wedge
the equality and estimate
left
function taken to the
of
R.D. CARMICHAEL AND R.S. PATHAK
418
[8, Lemma 2.2,
3.2], we have the
analysis on the exponentials and powers as in Carmichael and Hayashi
p.
70]
[8,
and Carmichael and Hayashi
proofs of Theorems 3.1 and
following important estimate for this paper"
(2ml+l)/4
<_ (I+K2)
IWk,m(pSt)I
F (ml-kl+(i/2)
exp(1Im21/2)ir(m_k+(i/2))l
Wkl,ml(PSlt)
(2,2)
SEPK
t>0.
THE WHITTAKER TRANSFORM FOR FUNCTIONS.
3.
Let k,m, and r be complex parameters, and let p and q be real parameters.
be a complex variable.
,m
(s)
p,q,r
Let s
The function
(st)r-(1/2)
Wk,m(pSt) f(t) dt
f(t) where __Wk,m(pSt) is
(3.1)
exp(-qst/2)
0
is the Whittaker transform of the function
function.
The general whittaker transform defined in
Srivastava
[9]
(3.1)
the Whittaker
was first considered by
for certain values of the parameters and variable which depend upon the
order of growth of
f(t).
In this section we prove initial and final value Abelian theorems for the
’whittaker transform defined in
(3.1).
For this purpose we now place conditions on
the parameters and variable noted above; these restrictions will hold throughout the
remainder of this section.
Re(m-k+(1/2)) >
0 and
q are positive.
The complex parameters k,m, and r satisfy
Re(r) > Re(m) >
Re(r) > (1/2). The real parameters p
that is Re(s) > 0.
in
0 with
The complex variable s is
>,
We now state and prove an initial value Abelian theorem.
THEOREM 3.1. Let q
ql+iq2 be complex with Re(q) ql > (-1).
f(t),
0 < t <
(e
for which
,
-ct
<_ t < and such that (f(t)/t )
(s) of (t)
Let the Whittaker transform Fk’m
p,q,r
If there is a complex number m for which
f(t))
@>.
is absolutely integrable over 0
< y for all y > 0.
f(t
Rim
t/0+
3.2
(:
tq
then for each fixed
K > 0
sq+l
im
PK
s e
Let
be a complex valued function such that there is a real number c > 0
is bounded on 0 < t
exist for s e
,m
(s)
(3.3)
A(q,k,m,p,q",)
where
A(D,k,m,p,q,r)
z+(z/2) F (q+m+r+l) r (q+r-m+l)
((p+q)/2 )q+m+r+l r (q+r-k+( 3/2
p
2Fl[n+m+r+l,m-k+(1/2)[
PROOF.
tq
and
Using Erdlyi et al [i0,
(st)r-(1/2)
exp(-qst/2)
(16), p. 216]
Wk,m(pSt)
dt
s
;q+r-k+(3/2)
we have
A(q,k,m,p,q,r).
C]Z
q+p
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
419
From (3.1) and (3.h) we obtain
sn+l ,m
(s)
p,q,r
o
s+
<_
I
1
+
a A(n,k,m,p,q,r)
()r-(l)
expC-qst/2)
(fCt)-at O)
Wk,m(pSt)
(3.5)
dt
12
where
I
s+]O+I
12 Is
y
(st) r-( I) exp(-qst/2)
Wk,m(pSt)
(f(t)-et n)
dtl
(st) r-(l12) exp(-qstl2)
Wk,mCpst)
(fCt)-t )
dt
0
Y
for y > 0 arbitrary for the moment.
(f(t)/t), (2.2),
I
sn+l
1
For s e
PK we
and estimates as in obtaining
y
I to
st)r-(i/2
exp(-qst/2)
0
use the boundedness hypothesis on
(2.2) to
obtain
Wk,m(pSt) ((f(t)Itn)-)
(nl+ml+rl+l)12
(3.7)
)/2)
sup
t <_ y
n I+I F (ml-kl+(l/2
I(f(t)/tn)-l
s
z
Ir(m-
k
(slt)rl-(1/2) expC-qslt/2) Wkl,ml(PSlt)
tl
dt
tit.
The integrand in the integral of this last estimate is positive valued for t
variable s
I
> 0 and the parameters
in the last estimate in
variable s
I
(3.7)
and r
Ol,p,q,kl,ml,
1.
by an integral over 0 < t <
> 0 and the parameters
and
Ol,p,q,kl,ml,
rl;
>_
0 for the
We thus replace the integral
and use
(3.h)
for the
by so doing the estimate
(3.7)
can be continued as
I
1
(rll+:ml+rl+l)/2
<_ (I+K2
ep(( Iql+ll
+lrl
r (ml-kl+(l/2)
sup
A(rll’kl’ml’P’q’rl) iI’(m-k+(172) )1
and this bound is independent of s e
12
given in
(3.6). By
PK
PK and holds
CfCt)/tn)
-
for each fixed y > 0.
we have
<_
t <
.
Putting e
-ct
into
12
and using
We now consider
-ct
f(t)) is
(e
(2.2) and esti-
hypothesis there is a real number c > 0 for which
absolutely integrable over 0
mates for s e
o<t<_y
R.D. CARMICHAEL AND R.S. PATHA
420
y
12 --I sn+l
<
(st) r-(1/2) exp(-(qs-2c)t/2)
Wk,m(pSt)(e-ct(f(t)-tn))
(+K)(n++r+)/ "((l_l+ll+lrl )/)
(slt)
rl-(l/2)
exp(-(qsl-2C)tl2)
nl+l P(ml-k+(i/2))
ir (-+ i )1
e-ct(f(t)-tO)
Wkl,ml(PSlt)
.
In this theorem we desire to prove (S.S) as
li
dt
PK"
+ (R),
As
tit.
sl
PK,
(R),
then
Re(s) /
Thus we now assume without loss of generality that
necessarily s
I
s
Re(s) > (2c/q) in the remainder of this proof for the fixed c > 0 and q > 0; and
I
for such s
Re(s) we know that
I
exp(-(qSl-2C)tl2) <_ l, t >_ O.
As t
(PSlt)
then
/
(3.10)
here mince p > 0 and s
I
> 0.
From properties of
(Whittaker and Watson Ill, Chapter 16]) including the growth at
Ko [6, line 2+, p. 351]) we have that
(ps It)
rI-(112)
Wkl,ml(PSlt)
function of t on y
(Pslt)
im
t/+
thus
(_t
rl-(l/2)
12 < (l+K2)
and
is positive, finite valued, and continuous as a
and
(
_
Wkl,ml(pslt)
[(PSlt) rl-(I/2) Wkl,ml(PSlt)
depending on s
Wkl,ml(PSlt)
(see also Tiwari
I such that Y
Oi
attains its maximum at some point t
tsl < .
Using this fact and
tsl
(3.10) we continue (3.9) as
(nl+ml+rl+l)/2
r (l-kl+(l/2)
"c
nl/l
P
-/(i12)
Ps
ltsl)
rl-(ll2)
WkI ,m1
Ps
Itsl
((t)
(3.11)
)
< (i+K
((121+11+121 )/2)
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
I’(m-z+Czl)
Ir(m
k
-rz+Czl)
(PY).nl.I
le-Ct(f(t)
mt)
+(z/a))l
Wkl ’ml (PSltsl
1+( i/2
)nl+r
(Psltsl
dt
0
where the last inequality is obtained from the fact that y
Re() > (-1) we
l
have
(te -ct)
<_
tsl <
is absolutely.lntegrable over 0
bined with the assumption in this theorem that
0 < t <
421
(e’Ctf(t))
(3.11)
yield that the integral in the estimate
and
i+i
> 0.
For
< t < ; this com-
is absolutely integrable over
From the hypothesis
is finite.
(3.2) and the estimate (3.8), which is independent of s e
given E/2 we can choose
y > 0 small enough so that I < /2. We now fix this y > 0 and it depends only on
1
+ +
then necessarily s
and as
s e
Re(s) +
E > O. As noted above, as
I
this happens
+
< =. Using the growth
since p > 0 is fixed and y <_
PK’
,
Isl
(Psltsl)
property at
tsl
of the Whittaker function
sl
smaller than
6/2
(3.5) yields
the desired conclusion
and
if
,
PK’
Wkl,ml,
(3.11)
is chosen large enough, s
PK"
shows that
12
can be made
Combining these facts with
(3.3). The proof is complete.
t/(l+t2), =l, c=l,
As an example for which Theorem 3.1 is applicable, let f(t)
t2/(l+t2), O l+i, c=l, and
u=l. Another example is obtained if we take f(t)
c:--O.
We now obtain a final value Abelian theorem for the Whittaker transform of
functions.
+
THEOREM 3.2. Let
t < =, be a complex
with
l > (-1). Let f(t), 0 <_-ct
valued function such that there is a real number c > 0 for which (e
f(t)) is abso-
l i2
<_t < and such that (f(t)/t 0) is bounded on y <_ t < for all
Let the Whittaker transform Fk’m (s) of f(t) exist for s e
If there is a
p,q,r
lutely integrable over 0
>.
y > O.
for which
complex number
f(t)
Aim
t++
(3.12)
t
then for each fixed K > 0
sn+Z,m
n
Isl/o
e
S
P’’
Is ml
Using
(3.7)
(3.)
Fk’mp,q,r(S)-
I and
fixed and s
as in
s"
(3.13)
PK
PROOF.
where I
(s)
12
and arguing as in
A(k,m,p,q,r) _< I +
I
12
have
(3.1h)
are defined in (3.6) for y >0 arbitrary. Let K >_ 0 be arbitrary but
Using the boundedness hypothesis on (f(t)/t), (2.2), and analysis
PK"
and
(3.5) we
(3.8)
we have
R.D. CARMICHAEL AND R.S. PATHAK
42
< (+K)
(+m+rl+l)/
and the right side of
exP((l21 +121 +1 =21 )/a)
F (ml-kl+( 1/2
(3.15)
is independent of s e
(3.15)
PK"
We now consider I
I
in
(3.6).
For the real number c > 0 in the hypothesis we argue as in (3.9) and obtain
I
1
2)
< (I+K
P
0
(PSlt)
exp(
where we have put p
there.
Isl
-rl+(1/2
rl-(1/2)
Ir(m
k
(3.16)
+(Z/2))I
le-Ct(f(.t)-tn)
Wkl ,ml (PSlt)
qsl-2c )t/2
dt
into the right side and hence have also put p
The desired conclusion
Isl
s
hl+l
r (ml-kl+( 1/2
(3.13)
-rl+(1/2)
in this theorem is to be obtained as
Re(s) 0 +; we thus may
I
< (l/p) for the fixed parameter
I
p > 0. As t / 0+, 0 <
< t / 0+ for all s < (l/p); by the growth condition at
I
0 in Tiwari and Ko [6, line h+, p. 351] or Whittaker and Watson [ll, Chapter 16] we
can choose constants M and T such that
/
0, s e
PK;
as
0, s e
PK’
then necessarily s
assume without loss of generality here that 0 < s
PSlt
Wkl ,m (PSIt)
Wkl
1
,m
I
(Pslt)
<_ M
(PSlt)(i/2)-ml,
(3.17)
0 <
PSlt
< t < T < y, s
I
< (l/p);
here M is independent of s < (l/p) and of t < T and of
t < t < T and T is indepenI
dent of s < (l/p). Returning to (3.16) and using
(3.17)
and
the fact that
I
<_ l, t > 0, we have
PSl
exp(-qslt/2)
I1 <_
(l+K2)(hl+ml+rl+1)/2 exp((In21+II+Ir21)/2)
s
hl+l
r (m-kl+(1/2))
1
Ir(m
+
T
(PSlt)
k
p
-rl+(1/2)
e
cy
(3.8)
M
(PSlY) rl-ml
+(/2))1
Wkl,ml(pslt)
e
-ct
(f(t)
le -ct
(f(t)-
t h)
Idt
0
I’
th)
dt
423
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
In (3.18), as in (3.11), we have
-ct
(e -ct f(t))
(f(t)-
t)i
rl-(i/)
interval T
;
is absolutely integrable over 0
absolutely integrable over 0
(PSlt)
dt <
!
t
maximum on T
!
!
t <
by assumption and
since c > 0 and
t <
Wkl ,ml(PSlt)
!
Re() > (-1).
(e -ct t o
is
Now
is a continuous function of t on the closed bounded
y for each fixed s l, 0 < s < (l/p); hence this product attains its
I
depending on s < (l/p). Continuing (3.18) we
t
y at a point
I
!
tsl
then have
I
I
<_
(i+) (ql+ml+rl+l)/2
)/2)
eCY si rll+l F(ml-kl+(1/2
r (m
k
!4
(PSlY)
rl-ml
P
(PSitsi)
+
)1
+(1/2
-ri+(i/2)
0 le-Ct(f(t)The estimate
(3.19)
holds for all s I, 0 < s <
I
(i/p).
rl-(1/2)
tn)
Wki,mi(psitsi)
dt.
The constants
c,K, and M are
independent of s I, 0 < s < (i/p), as are all of the parameters p, k
k + ik2,
I
I
m
m + im2, r
+
q
r
ir and
+
and the yet to be chosen constant y > 0.
iq2
ql
I
I
2
Again as
then s
Re(s)
0, s a
0+. Recall that T <_
<_y for
sl
PK’
sl
0, s
PK’
T’
Sl,
which are independent of s
Wki,ml(Psitsi)
I
0 < s
Psltsl
the growth condition Tiwari and Ko
Whittaker and Watson [ii, Chapter 16] applied to
and
tsl
I
< (I/p) in (3.19) and T is independent of
I
>
on y
0.
(Recall (3.1).) As s I O+ then 0 <
0 < s
[6,
I
< (i/p), but dependent
<_PSlY
line
+,
Wkl,ml(PSltsl
0+.
Thus as
p. 351] and
yields constants M’
such that
Wki,mi(PSitsi) ! M’
(PSitsi)(i/2)-mi
(3.20)
0
for 0 < s
I
I
< (l/p).
with the estimate
1
proof of Theorem 3.1.
Using
(3.20)
(3.15) on
12,
in
(3.19)
<
Psltsl ! PslY < T’
and combining the resulting estimate on
the proof can be completed similarly as in the
Examples of the applicability of Theorem 3.2 similar to those after the proof
of Theorem
3.1 for Theorem 3.1 can be constructed by the reader.
R.D. CARMICHAEL AND R.S. PATHAK
424
4.
THE WHITTAKER TRANSFORM FOR DISTRIBUTIONS.
We construct a Whittaker transform for distributions as in Pathak [12]. We
define the differential operators D
as in Paths/< [12 (3), p. 4];
0,1,2,
the seminorms
[12,
Pathak
#
Let s
exp(-pst/2)
PROOF.
(0,) are
given in
6].
LEMMA 4.1.
((st)re-(i/2)
and the test functions V
0,1,2,
_Ya,
p.
Re(s) > (a/p).
0 and
Let Re(m) > 0.
Wk,m(pSt)) e Va(0,)
(0).
as a function of t e
[12, Lemma
The proof is by Pathak
We have
l, p. 6].
Nothing different is intro-
Wk,m(pSt)
duced by having p > 0 in the exponential term and in
here.
Let V (0, ) denote the space of generalized functions defined on V
[12 p. 6]).
Let U
V
Va(0,)
that is U e
(0,).
a
Let
for each a >
U
U
have the meaning given in Pathak
Va(0,)
ff
and U
for any a <
U"
(pst)>
U
(0, ) (PathOs:
[12 P. 7]"
Because of
Lemma 14.1 we can form
wTk’m[u;s]
p
where
U
<U
t
{s:s#O
(st) m-(I/2) exp(-pst/2)
Re(s) >
is an analytic function of s in
The set of seminorms
s e
< Arg(s) < ’} and Re(m) > O.
qU’
V
the distributional Whittaker transform of U
WTk’m[u;s]
P
Wk, m
{a}
a
(0,,);
(14.1)
We call
wTk’m[u;s]
p
by Pathak [12 Theorem l, p. g]
fU"
which defines and generates the
01,2
topology of V (0) is a mu_itinorm in the sense of 7.emanian [13 p. 8] as noted in
a
is a norm. nus the hypotheses of Ze.n+/-an [13 Theorem
Pathak [12 p. 6] here
Y0
1.8-1
p.
constant
18]
are satisfied; by this result, given U
and a nonnegative integer
<
u,
>
N
, v()’
c :o,,
<
V
(0 ’) there
U such
which depend only on
exist a positive
that
(.)
va(o,).
We shall call the number N here the order of the generalized function U e
Va(0,).
We obtain Abelian theorem. for the distributional Whitter transform; in so
doing we use the results of section 3. Thus in the remainder of this section we assume
that the complex parameters k and m satisfy Re(m-k + (1/2)) > 0. By comparing the forms
(3.1)
and
(4.1)
we see that the complex
and section 3 are now taken to be r
the restriction
Re(m) > (i/2)
parameter r and the real parameter q of (3.1)
m and q
p in this section.
We thus must make
in the remainder of this section because
Re(r) was
necessarily so restricted in section 3.
wTk’m[u;s]
P
To prove our initial value theorem for
LEMMA 4.2.
T
it <
,
B > 0 and
Let U e
T > 0.
Y0
Va(0,)
Let s e
PK
for a >
with
U
we need the following lemma.
and let the support of U be in
(p Re(s)) > max {l, p GU, 4a).
There are constants
> 0 which are independent of s such that
wTkm[u; s]
<_B (i +
where N is the order of U.
s
I)
m+kl+N-
1/2
exp(-PSlY0/h),
s
I
Re(s),
(4.3)
ABELIAN THEORERS FOR WHITTAKER TRANSFORMS
The proof of this lemma is in the spirit of that of Zemanian
PROOF.
p. 2h6].
425
(t) e C
Choose a function
[2, Lemma i,
such that for any nonnegative integer
d(k(t)
m we have
-<t <,
dt
is a constant which depends only on a; 0 <
where M
k(t),
and the support of
supp(k), is
<U, > <U,
denoted
Then by the usual proof we have
<U, l>
and the value of <U,$>
Let Re(s) >
above properties.
w;m[u; s]
a
for
Re(s) >
N
0 < t
>,
eat D(kCt)
[
exp(-pstl2)Wk, m
(st )m-( 112
fJ(t)’
t8
As
8=0
constants.
complex
are
where the A
8
of the derivatives of k(t)
(2..17), p. 25] we continue (.) as
sup
max
0 < t <
,N
d
Using
s
k(t)
Ii
eat
(st)m-( i/2)
(h.5),
,N 0 < t <
1
d
t8
dt6,N
p
[
8- (.l (t)
max
1
[T,);
the
we apply
(pst) >
(.2)
expC-pst/2)
!
Wk,m(pSt))
(I.5)
the Leibnitz rule, the boundedness
in (4.6) below), and Slater [lh;
sup
0 < t <
dt
e
8=0
-m*(112) (ps) (-i ) (pst
Ag
(st
m
at
%t
.
eat
sup
max
< C
a=0
Va(0,)
._
exp(-pst/2)
dt 8,
p:A=(l’-i0!
< T.
k(t) satisfying
<_
< C
=0,i,
8
Y0
supp(U)
0 1,2
(IdB-(k(t))/dtS-61< MS,
m=0
0 <
;
of the seminorms to obtain
X(t) (st) m-(I/2)
<
m (f(t))
=C
since
i for T < t <
From Pathak [12, (8), p. 5] we have
GU"
w[us]l
Y0 ! t
Va(0,),
e
With N being the order of U e
GU"
sup
max
0,i,
<--C
<Ut,
contained in
< ,
is independent of the choice of
[12, (12), p. 6]
and the definition Pathak
l(t) < I; k(t)
t
(pst)I
g
)m-(/2)
I
Wk,m
exp(-pst/2)
Wk, m
81
:
8 0
)m-(112)-(12)
--(s)-,
exp(-pst/2)
(pst)
II
R.D. CARMICHAEL AND R.S. PATHAK
426
Wk+C12),m-(12)(pst)
C
,N
[
0 < t <"
8’.
8
sup
max
a=0, l,
A
B=c
8
MS, 6
6=0
Jsl-S le at (pst)m-Cll2)-(12) +8 exp(-pst/2)
p-m+(ll2)+-S
Wk+CSl2),m_C12)(pst)
PK
in the
4a}, and s e
Re(m) >_ (1/2),
p Re(s)) > max {i, poU
1
hypothesis of this lemma; under these restrictions we use the growth properties of the
Now recall m
[ii, Chapter 16] and Tiwari and Ko [6, p. 351])
Whittaker function (Whittaker and Watson
and analysis as in the proof of Pathak
[12, Lemma 2, pp. 7-8] to
obtain a constant C
8
such that
e
(pst)m-(1/2)-(6/2)+8
at
exp(-pst/2)
<_
Wk+(6/2),m_(6/2)(pst)l
C
8
Ip-m+(i/2)+6-81
texan Isl -8,
p
-m+(/)+-S
0,i,
,
S
exp(-PSlt/4)
(l+s I)
holds for t > 0 and all relevant values of the parameters.
<
Now in (4.6) we notice that
the
(p Re(s)) > max {i, lU,ha} and s g
,c, in (h.6) is bounded by a constant which
supp(k)
[y0 ,) with 0 < YO < T so that the sup in
PK’
Since
0, i,
.
and 8. Now recall that
Hence for the values of t actually
(h.6) is actually taken over Y0 < t <
with the
being considered in (h.h) and (h.6), the estimate in (h.7) holds for Y0 < t <
(h.6)
and
(h.7)
with
term exp(-PSlt/h) replaced by exp(-PSlY0/h). Combining these facts
depends on
(h.h)
and
we obtain a constant B which depends on the generalized function U and on its order N
such that the
s.
desired conclusion
(.3) holds;
the constants B and
Y0
are independent of
The proof is complete.
We now prove an initial value Abelian theorem for the distributional Whittaker
(0,)
Let U e V (0, ) for
a
transform where the element U e V
is assumed to have support in
[0,).
a >
THEOREM h.l.
U such that over some right neighborhood
(0, t 0) of zero U is a regular distribution corresponding to a complex valued function
-ct
f(t)) is absolutely intef(t) such that there is a real number c > 0 for which (e
For
grable over 0 <_ t <_ t
i+i2, i > -1, let (f(t)/t D) be bounded on 0 < t
O
exist for the
s), s e
Let the Whittaker transform
< y’ for all y’ < t
p
O
for all y’ < t
function which is f(t) on 0 < t < y’ and which is zero on y’ <_ t <
O
Fa’_m,m
(,
If there is a complex number
f(t)
t0+
t
>,
for which
(.8)
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
then for each fixed K > 0
s+l WTk,m[ U ;s
im
c,.
seP
K
( ,k,m,p, p ,m)
We decompose U into U
PROOF.
0 < T < t
O
I)
[0,T]
and
supp(U2)
m
tU1;s] + wTk,p [u2;s].
WTk,m"
p
Lemma h.2 is applicable to WTk’m
Isl
2
(h.9)
[T,=),
We now have
wTk’m[u;s
p
,
+ U where supp(U
1
_
427
p
s e
PK’
[U2;s]
then necessarily s
(4.1o)
since
supp(U2)
.
Re(s)
I
C
[T,), T >
0.
Recall that as
Thus by the conclusion
(4.3)
of
Lemma 4.2 and the estimate
sn+l ! (+K) (nl+l)/2 (s) nl+l exp(ln21/2)
s s
PK’
we have
s n+l
im
Extend the function
0<t<
f(t)
(t)
o.
in the hypothesis, which is known on 0 < t <
to,
to
by
(t)
Then
s
A(h,k,m,p,p,m)
RE
s e
wTkm[u2
=If(t)’
0 < t < T
0, Tit <.
satisfies the hypotheses of Theorem 3.1 and
wTkm[ul;s] wTk;m[(t);s]
with this latter Whittaker transform equaling the function Whittaker transform of
(t)
defined in
(3.1).
Zim
Combining
Thus by Theorem 3.1
s +I
(4.10), (4.11),
wTk’m[uI ;s
and
(4.12) we
have the desired result
(4.9).
The proof is
complete.
COROLLARY 4.1.
hood
(0,t 0)
Let U
V’a(0’ )
.
U
such that over some right neighbor-
of zero U is a regular distribution corresponding to a
which is Lebesgue integrable over 0
(f(t)/t )
for a >
<_ t
<
Let D
nl+i 2, i
be bounded on 0 < t < y for all y > 0 and let
complex number a.
Then for each fixed K > 0
(4.9)
(4.8)
holds.
function f(t)
Let
> (-I).
be satisfied for some
R.D. CARMICHAEL AND R.S. PATHAK
428
PROOF.
Since
f(t)
is Lebesgue integrable over 0
absolutely integrable over 0
<_
<_
t <
for any fixed c > 0.
t <
then
(e -ct f(t))
is
_
The result thus follows
immediately by Theorem
We now proceed to obtain a final value Abelian theorem for the distributional
Whittaker transform.
First we need to make some comments concerning this transform.
Let
of a distribution of the form assumed in the final value theorem below.
oU, with supp(U) [t O ,), t O > O. Assume over some interval
with 0 < t < y that U is a regular distribution corresponding to a complex
y < t <
o
U + U2 with supp(U
valued function f(t). Then U can be decomposed as U
1
[T,), T > y. We then have
and supp(U2)
U e V
(0,),
a
a >
I)
WTk’m [U;s]
P
WTk’m
P
[Ul;S]
+ WTk’m
P
_
[t0,T1
(h.13)
[U2;s].
in which Re(s) > O
From the definition (h.l) WTk’m [U;s] is defined for s e
U
P
V (0, ) assumed above and the assumptions on f(t)
Because of the form of the U
a
h.2 below, where we use the above decomposition, we can
Re(s) > 0.
here, and (h.13) will be well defined in Theorem h.2 for s I
in the final value Theorem
0
take G
U
Because of this we may let
sl0,
PK’
s
desired.
n
U
in the final value theorem below as
We now obtain a needed lemma for the final value result.
0 < t < T
with supp(U)
LEMMA h.3. Let U e
O
0
with n > -i. For each fixed K >_
I
i
’
+i2
im
i1.o
,
n/
[t0,T],
[u;s]
Let
o.
seP
K
The distributional Whittaker transform
PROOF.
,m
P
[Us]
of U
’
Re(s) >
exists for
N
U=
[
0
By Schwartz [15
Th&orBme, p 91]
d((t))
=0
dt
g(t),
for some nonnegative integer N, the order of U, where the
e
0,i,
O
distributional differentation and the calculation Slater
have
WTk’m [U;s]
P
[
6=0
(st
g(t)
(-i)
to
dt
,N,
[t O -g, T +g ],
-g > O. Using
O
[lh, (2..17), p. 25] we
are continuous functions with support in an arbitrary neighborhood
T]. Since g> 0 is arbitrary we assume here that t
> 0, of [t
exp(-pst/2)
Wk,m(pSt)
dt
ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS
N
T+@
I
t0-E
)e
=0
ga( t
P
-m+(i/2)
(ps)
expC-pstl2)
Now each
g(t)
is continuous on t
Iga(t)l
<_ Ma,
for constants M
a
t
(
O
!
o
<_
t
,N.
0,i,
<_
t
(_l)(pst)m-(i/2)-(/2)
Wk+(al2),m_Co12)(pst)
T +
!
hence
T + ([
(4.16)
,N,
0,i
For each a
dt.
,N we have
0,i,
Is+l <_ <+) (nl+l+)/2 ep(lll) s Ii+i+
s e
Using analysis like that in obtaining the estimate
(4.7)
PK’
Re(D)
rll
>-i.
in the proof of Lemma
(4.17)
4.2
we have
(pst)m-(i/2)-(cx’/2)
Wk+(a/2),m_(c12) (pst)l
exp(-pst/2)
<_
(4.18)
< C
for constants
Isl
0, s e
(4.17),
and
with a
%
PK’
,N and s e
0,i,
then necessarily s
(4.18)
Re(s)
I
(4.14)
we obtain
(l+s I)
0
ml+kl-(i/2
PK"+.
exp(-Pslt/4)
Again recall that as
Thus by combining
(4.15), (h.16),
and the proof is complete.
We now obtain a final value Abelian theorem for the distributional Whittaker
trans form.
Va(0,),
a > OU, with supp(U) _C
t > 0. Over some
O
< y, let U be a regular distribution corresponding to a
O
complex valued function f(t) for which there is a real number c > 0 such that
-ct
e
> (-1),
+
For
f(t)) is absolutely integrable over y <_ t <
THEOREM 4.2.
interval y < t <
,
Let U e
[t0,)
0 < t
.
let
f(t)/t
be bounded on
Fk’m
(s), s e
p,p,m
zero on 0
<_
t
>,
<_y’
y’ <_ t <
for all
y’
>_ y;
exists for the function which is
for all
y’ >_ y.
im
t++
i2’ hl
l
and assume that
f(t)
on y’ < t <
If there is a complex number
and which is
for which
f(t
a
t
(4.19)
then for each fixed K > 0
im
s rl+l WT
k,m
p
[u;s]
(4.20)
se
PK
A(,k,m,p,p,m)
R.D. CARMICHAEL AND R.S. PATHAK
430
[t,), T > y,
l)
and by the discussion in that paragraph,
function
f(t)
[t0,T]
and
U + U2 with supp(U
1
as in the paragraph above which contains equation
Decompose U into U
PROOF.
supp(U 2)
(h.13)
holds for s
in the hypothesis, which is known on y < t <
Re(s) >
(h.13);
Let the
0.
,I
be extended to
P
[U2;s]
O<t<by
(t)
Then
(t)
{0
0 <t <T,
f(t), T< t<
satisfies the hypotheses of Theorem 3.2.
Now
P
[(t);s]
with this latter Whittaker transform equaling the function Whittaker transform of
defined in
(3.1).
Theorem 3.2.
Thus the conclusion
(h.20)
follows from
(h.13), Lemma h.3,
(t)
and
The proof is complete.
In future work we hope to extend the Abelian theorems of Joshi and Saxena [16]
and Malgonde and Saxena [17] for the H-transform to the general setting that the
complex variable of the transform approaches 0 or
inside a wedge region in the right
To do so we will need to use the proper-
half plane and for more general parameters,
ties of the H-function as in Srivastava et al
[18].
Analysis which is associated with
that in this paper and with the corresponding H-transform problem is contained in
Sinha
[19]
and Joshi and Saxena
[20].
The
MeiJer transform is studied in Pathak [21]
MeiJer and Whittaker transforms.
which we note here because of the similarity of the
5.
ACKNOWLEDGEMENTS.
In a letter to one of us (R.D.C.) Professor H. M. Srivastava has made some
We thank Professor
comments concerning certain analysis contained in this paper.
Srivastava for this contribution to our paper.
One of us (R.D.C.) expresses his sincere appreciation to the Department of
Mathematical Sciences of New Mexico State University for the opportunity of serving
as Visiting Professor during
1984-1985
when the research on this paper began.
The research of Richard D. Carmichael in this paper is based upon work supported
by the National Science Foundation under Grant No.
DMS-8418h35.
Some of the research of R. S. Pathak in this paper is based upon work which was
partially supported by a grant from the William C. Archie Fund for Faculty Excellence
of Wake Forest University.
This author expresses his appreciation for this support
while visiting Wake Forest University.
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