671
Internat. J. Math. & Math. Sci.
No. 4 (1987) 671-692
Vol. I0
A GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS
JACOBI TRANSFORM
AHMED I. ZAYED
Department of Mathematics
California Polytechnic State University
San Luis Obispo, CA 93407
U.S.A.
(ReceLved August 19, 1986 and in revised form
ABSTRACT.
September I0, 1986)
In this paper we extend the definition of the continuous Jacobi
transform to a class of generalized functions and obtain a generalized inversion
formula for it. As a by-product of our technique we obtain a necessary and
o to be the
sufficient condition for an analytic function F(X) in Re
continuous Jacobi transform of a generalized function.
KEY WORDS AND PHRASES.
Continuous Jacobi transform, Generalized functions,
Inverse Jacobi transform.
1980 AMS SUBJECT CLASSIFICATION CODE. 44A15, 33A65 and 33A3
I. Introduction:
The ,lain principle in sampling theory is to reconstruct a function f(t) defined
on some open interval
sequence of points
from the knowledge of its values
{tn}n=O2
f(tn)
at a discrete
I. In Kramer’s sampling theorem [9 ], this can be
expressed in the form
tI
] f(t n )S n (t)
(l 1)
n= 0
where {Sn(t)}n__ 0 is a sequence of determined functions.
Suppose that
is a complete system of orthonormal functions defined
f(t)
{,n(t)}n=O
on
with respect to some measure
do(t). Let us call these functions p-functions.
In the case when f(t) can be expanded in terms of the p-functions,i.e.,
a(n)
f(t)
n=O
where
determining
a(n)
{Sn(t)} 0
/
f(t)
(1.2)
Cn(t)
@n(t)
dp(t)
(I.3)
requires generalizing the p-functions to the case where
672
A.I. ZAYED
the discrete parameter n is replaced by a continuous real parameter
in some
open interval J.
This has motivated the idea of generalizing various classical
orthonormal
sequences of functions to the continuous parameter case which in turn has
led
to the investigation of expanding functions in terms of the generalized
p-functions
{@(t)}j
If we call
(I.)
the discrete p-transform of
f(t) and
(1.2) as its inversion formula, then analogously we can define the
continuous p-transform of f(t) as
regard
ff(t) (t)dp(t)
a()
But since the
q>,(t)’s
(1.4)
are no longer orthogonal, finding an inversion formula
for the continuous p-transform
(I.4) is usually more difficult.
An explicit ,upese,tatio, u {Sn(t)) has been given in [2 ],[ 7] and [ 8 ]
for the cases where the p-functions are the Bessel’s functions, Legendre, associated
Legendre,Gegenbauer,Chebyshev,Laguerre polynomials. However, only in a very few
cases an inversion formula for the continuous transform is known. Inversion
formulas for the continuous Legendre and the continuous Jacobi transforms were
derived by Butzer, Stens, Wehrens
and Deeba, Koh [5. ] respectively.
[1
It should be pointed out that Debnath [3,4] was the first to
introduce the theory of the Jacobi transform and discuss the
operational properties of this transform with applications.
The main purpose of this paper is to derive a more general inversion formula
for the continuous Jacobi transform which will include the results of [1 ] and
[ _s] as a special case. As a by-product of our approach we
obtain a more general
definition of the continuous Jacobi transform which enables us to give a
necessary and sufficient condition for an analytic function F() in Re
>
0
to be the continuous Jacobi transform of a generalized function f(t).
2.
Preliminaries:
In this section we recall some of the basic background material necessary
for our investigation.
For any real numbers a,
function
F(a,b;c;z)
b
and
2Fl(a,b;c;z)
c
with
c # 0,-I,-2,
is given by
the hypergeometric
.
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
F(a,b;c;z)
where the series converges at
c
and
a
b > 0
p(,B)
(x)
X
,S >
N:o-T kk k-’T- z
-I
z
and
k
Izl
z
<
(2.1)
provided that
c
a
b +
> 0
respectively.
The Jacobi function
where
(a)k(b) k
673
P(z’B)(x)
of the first kind is defined by
F(+e+l
r((z+l)r(+i) F(->,,)+(Z+B+I ;(z+l
] ),
-I, >, EII and ,+(z+l # 0,-I,-2
p_,B)Cx
r(-+l)r(--B)
r(1-)r(-s)
we may restrict ourselves to the case
)
x
(-I,I]
(2.2)
Since
p((z,B)
--s- (x)
>_.-((z+ 2B+]).
The function
P)(’B)(x)
satisfies the differential equation
(l-x2)y + (B-c-((z+B+2)x)y + )(},+(z+B+l)y
O.
(2.3)
Let
p(z,B)(x) (l-x)(l+x)BP’B)(x)
(2.4)
and
t
(l-x2) d--# + [((z-B) + (o+B-2)x]
Then with some easy computations,one can show that
of
L with eigenvalues
-},(,+m+B+l),
+ ((+B).
(2.5)
P’B)(x) are the eigenfunctions
i.e.
(2.6)
One can also verify that the conjugate differential operator
t
.
2
d
(l-x2) xB-f
[((z-B) +
d
((z+B+2)x]-
L*
of
L
is given by
(2.7)
A.T. ZAYED
674
-I<
,
-(+2B+l),
that for },,v >_
It can be shown
>,
v,
# -(++B+l)
},
and
B
(l-x)(l+x)BP’B) (x)P (B’) (-x) dx
{ r(v+l)](,++
I’(>,++l )r(v+B+l
sin v
sin
-(>,-v)(>,+v++B+l
If we denote the Jacobi polynomial of degree
2+B+l
n
Pn(’B) (x)
by
(l-x)(l+x)Bp ,B)(x)p(’B)(x)dx
m
-l
}’
anm h n
it follows that
(l_x)(l+x)Bp
2+B+l
(2.9)
(2.o)
n! (2n++(+ r (n+e+B+l-)-
Pn(’B)(-x) (-1)npB’)(x)
(2.8)
then
r(n++l )r(n++l
hn
Since
i(+I )r(v++t+l
+i)
-1
,B) (x)p(,B)(x)dx
n
=
(-I)nr (++I) r(n+B+1 )sin
--(},-n)(),+n++B+l)’n!r(>,+=+B+l)
)’
# n
(2.11)
(-l I]
(2.]2)
and hence
p
,B)(x)
,B)
n--O
(n)Pn(,B)(x)
x
The following estimates will also be needed:
i) for -l < B
ii) for B > O,
iii)
<
{
l(l+x)Cp ’B) (x)l
x[-I ,I]
O,
sup
iv) as a function in
polynomial growth as
K:())
as X
/(R)
(R).
(
=,
for any c > 0
unifomly in x c [a,b]
the supremum K()
,
for any c > 0
(2.13)
{
I(l+x)B+Cp’B)(x)l
x[-l,l]
sup
P’B)(x) 0(-)
<.,
in
c
(2.14)
(-l,l)
(2.13) and (2.14) is at most of
In fact,
0(, q) as I
where q
max(.B).
(2.]6)
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
and the action of
Let
x
f c
(a,b),
on any member
-< a < b
q e X v:ill be denoted by
<_oo and C(1)
I.
different iable functions on
f,
be tIe class of all infinitely
The space of all infinitely differentiable
provided with its standard topology as described in [lO]
functions on
E(1) while P(1) will denote the space of all functions
will be denoted by
C’(1) whose support
topology of P(1)
is its standard topology as described in
(m,B) (n)
and the series expansion of
n:O
2+B+l
-I< B <
(,B)()
-l
It can be shown that for -I
L2{(-I,I)
f(x)
w(x)} and
4
].
p > I.
Then the
is defined by
(2.17)
-l
nnl (,B)(n)p(,B)n
continuous Jacobi transform
(,B) (>,)
f(x)
of
B
f(x) in terms of the Jacobi polynomials is given by
>-I,
Analogously, if
LP{(-l,l),w(x)},
[
The
(l-x)m(l+x)BP ’B)(x) f(x) dx
2m+B+l
f(x)
f(x) e
and
(m’B)(n)
discrete Jacobi transform
I.
is compact and lies entirely inside
(l-x)m(l+x) B
Let w(x)
f(x) e
X*
X denotes a locally convex space, then its dual will be denoted by
If
@(x)
675
(x), x
f(x)
and
of
(-l,l).
(2.18)
L2{(-l,l),w(x)}
f(x) will be defined by
(l-x)(l+x)BP’B)(X) f(x)dx;
< m,
the
-I < B
V-(m,B)(,-1/2)
m+B
and
<
e
L I(0,)
JO(e B)(,_1/2).P((’()(-X) HO()},
-1/2
sin
}, >
0
-1/2.
(2.19)
if
then
(2.20)
almost everywhere in (ol,l) where
Ho(k)
r2(x+1/2)
r(x++1/2)r(x+s+1/2)
(2.21)
A.
676
.
ZAYED
is continuous on (-l,l),then (2.20) will hold everywhere
In addition,if f(x)
on (-l,l).
Here we should point out that in [ 5], formulas (2.19) and (2.20) were proved
for
-I
1/2 <
<
B
,B
however they can be easily extended to the case where
2
since
is integrable over (-l,l).
<
Similar argument can be applied to
3.
IP’B)(x)I
(l+x)B(l-x)
<
(2.8).
J_aco__b_i_ transform of a class of generalized functions:
will denote the open interval (-l,l).
Throughout the rest of this paper
Fix
and B
such that ,B > -l
We define the space
of all infinitely differentiable functions
sup
Yk,c () xI
(x)
as the space
/Y
defined on
J(l-x)-(l+x)Ctk(x)J
such that
(3.1)
<
for any nonnegative integer k and any c such that max(-B,O)< c <
Clearly,
,B
is a linear space under the pointwise addition of functions
The collection I"
and multiplication by complex numbers.
is a separating family of seminorms since each
seminorm and
by the family
YO,c
is a norm.
In fact
The topology of
Yk,c (k
H,B
{Yk,c}k=O,max(B,O)
1,2
is the topology induced
it is not hard to show that the same topology can
be generated by a countable subset of
.
Provided with this
becomes a locally convex, Hausdorff topological vector space.
argument similar to the one used in lemma 3o2-I in
,
H
H, B
topology,
H, B
By using an
B], one can show that
is in fact,a Fr{chet space and in addition to that
where the topology of
is a
P(I)
H, B
E(1)
is stronger than the topology it inherits from
E(1) and likewise the topology of P(1)
is stronger than the topology it
<c <l
GENERALIZED INVF.SION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
inherits from
Consequently, the inclusion
/Ve,B.
Moreover, if f(x)
f(x)
l) for any. l> c> max(-B 0)
<f’>
for any
,
H, B
/Y*,B
D
is a locally integrable function on
Ll(-I
(l-x)-(l+x) c
I)*(1)
since by
I<f,>l
<
I
E*(1)
holds.
such that
iv*,B
f(x)
and
f(x)@(X)dx
-I
(3.1) for k
0,
we have
(I-x)
(l-x) -( l+x )c
-I
then
D
677
]+x)
(x)dx
<AI ;-(Xl )cldX<
(3.2)
+x
-l (l-x
for some positive constant A.
Evidently ,L
is a continuous linear operator from
Yk,c (L)
Thus
xSUpc (1-x)-=(l+x)CLk(L)
L*
it is easy to show that the operator
into itself since
(3.3)
Yk+l ’c()"
defined on
by
(3.4)
into itself.
is a continuous linear operator mapping
Moreover, for any fixed
a nonnegative integer
f e
(*,B’ there
p such that for any
I<,,>I <_c
Since
exists a positive constant
0<k<p
C and
e Y
y (,).
(l_x)(l+x)Bp.(,B)(x)I
p.,B)(x)l
A
is a member of
H
as can be
seen from (3.1), (2.13) and (2.14), we can define the continuous Jacobi transform
F(’B)(,)
of any member
f(x) of the space iV*
F (’B) (},)
by
(x).
2+B+l <f(x) ,p,B)
A.I. ZAYED
678
Formula (3.6) reduces to (2.19) if
f(x)(l-x)
(l+x) c
such that
on
L (-l,l)
f(x)
is a locally integrable function
for any
c
with
max(-B,O)
< c <
I.
Some properties of the continuous Jacobi transform will be given in the
following proposition.
But first we need the following lemma"
Lemma 3.1
a.
(.d_[pm,B)(x))
Proof"
P(m’B)(x)
Since
is analytic in
H,B
Re
in the region
> 0
Re
> 0
it suffices
to show that
Yk,c
for
(3.7)
P
O,l ,2
k
We show this by using induction on
k.
First we show that
(3.8)
i)
if
-l < B < O,
where W(x)
we write
(l-x)(l+x) B
G(,)
I’(+l)r(>,+l)
F(-;k,>,++B+l;+l,]-)
an(,,(3)(]-)n
and
an(’’B)
r(-+n) r(++B+l +n) r (+I)
r(--)r(++s+l)r(+l+n)r(n+i)
(3.9)
Hence
d pCe,B)
d
W(x)
G’()F(-,,++B+I;+I,
+ G()
n=O
an( ,,(3) (]--) n
}.
(3.10)
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
F(-,A++B+I
Since
;+l,)
YO
is defined and bounded on
(W(x)F(- ++B+I ;+I ;-))
679
[-l,l],then
<
(3.1l)
By taking the logarithmic derivative of (3.9) we obtain
n=O
n
+
I’()++B+I>
’I’{$B+!
F(-,++B+I ;+l,
i’(++B+l+n)
0
+
12 +
.I’_’ (X+:/B+l+n)
Since
(3.i2)
O(-I_B)
n
as
is uniformly and absolutely convergent on [-l,l] and hence
the series defining
Yo,c(W(x))
12
an(X,,B)
and
(j_)n
is treated similarly.
<
-.
The last term on the right-hand side of (3.12) has
already been shown to satisfy (3.11).
Thus by combining this, (3.10) and (3.11)
we obtain that
YO’ct
i)
if
B
O,
dp,B)
d.
one can show, by using the same argument as in (i), that both
F(-,,)++l;(x+l;)
and
d F(->,,),++l;+l;]-)
are O(n
l-)
as
x/
-I,
hence they satisfy (3.11) and consequently (3.8) holds.
iii)
if
0 < 8,
we use the following contiguous relation for the hypergeometric
function
F(a,b;c;z)
to write
(l-z) c-a-b F(c-a,b-a;c;z)
A.I. ZAYED
680
The proof is exactly the same as in part
F(++I,-;E-B;+I;])
(i) since in the series expansion
a
n=O n
O(nl)
an(,,B)
(,,,B)(-) n
as n
/oo.
Finally, from (2.6) we have
(3.]3)
where
n()
-(++B+I ).
(3.14)
Therefore,
and hence
The last inequality follows from (3.8) and the fact that
Proposition 3.1:
and
(x)
Let f(x)
H*
F
P’B)(x) c Ha,B.
Q.E.D.
be its continuous Jacobi transform
(’B)()
P(1). Then
i)
F(’B)(},)
ii)
F(’B)(X)
0(>, q) for some q > 0 as
is analytic in
},
Re X > 0 and
/-.
dF(’B)(X)dX +B+II <f(x),ld p,B)(x)>.
2
iii) the continuous dacobi
(=’B) (X)=
is rapidly decreasing as
transform
(,B)(,)
of (x)
2+IB+l [-I (x)(l-x)=(l+x)
},
(R),
i.e.,
(’B)(;k)
B
given by
P=’B)(x) dx
O(;E -p) for any p > 0 as
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI
TRANSFORM
Proof-
681
i) In view of (3.5),there exists a constant C and a nonnegative
integer
p such that
IF(e’B)()I < 2--T-i-C max
O<_.k<__p
sup
x
l(l-x)-(l+x) c
Lk p ,B)(x)l
(3.15)
By combining (3.15) and (2.16) we obtain
IF(’B)()I
<_ B
where
K()
and
for
B
q
c [In(x)l] p
<_
#(-x)’(+K) c
sup
[ln(x)l]P
is a positive constant independent of
0(, q’) as
2p + q’.
/,=o
where
q’
i) With some easy computations one can
F (’B) (,+A),)
P’B)(x)#
But since n(},)
0(2)
max(,B), it follows that (i) holds
show
that
F(e,B)
(,) (x )>
2+B+i
2+B+I <(x),%x(x,x)>
(3.16)
where
(,B)
p(,B)(x d
ax(x’x) :’+x’x)
--BTP
By lemma 3.1,
d- P’B)(x)
Nm, B,
belongs to
? B) (x).
hence (3.16) makes sense.
T)ie proof will be completed if we can show that
lim
in
6(
(x,},)
(3.17)
0
Again with some easy computations one can show that
(I -x)-(l+x) C
LkA), (x,)
(l-x)-(l+x) c
{
[n(+ap)]k p(,B)(x)
"+A
[q(),)]k p ,B)(x)
L
A.
682
But since
[.(,I)] k
Pa’B)(x)
.
ZAYED
is analytic in
it can be expressed as a Cauchy
>,
integral and thus we can write (3.18) in the form
(l_x)(1+x) c L k
where I"
is a closed contour encircling the points
inside the region
Re
’B
Let
> 0.
sup
xcl
scF
then, from (3.19) we obtain
(l_x)-O(l+x) c L k
where A
and
d
as A
iii)
2
/y
-
(3.19)
d-,
is the length of
distance(>,+Ak,I’).
CAx(x,X)I
F
and d
),,
l(l-x)-C(l+x)C[n(s)] k Ps(’B)(x)I
<_ C’BA(A),
k
min(dl,d2),
From (3.13) we have
(x)
2+B+i -I
2+B
+l
2+B+l
[n(X)]
(,B) (x) dx
Px
(x)[n()’)]k PX(,B) (x)
k
[n(),)]k
for any nonnegative integer
(3.20)
d
distance(,I’)
But the right-hand side of (3.20) approaches zero
and this proves (3.17).
(e’B) (X)
and lying entirely
X+Ak
k.
-I
P’B) (x) (L*) k (x) dx
Thus,
dx
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
%(a,B)(),) O(),-2k)
it follows that
But since n()k)
by (2.15).
for some suitable constant c
as
O(X 2) as X
683
/
(R),
for any nonnegative integer k.
),
Q.E.D.
Let f(x) e
Lemma 3.2"
and
p(1).
(x)
.
(),) be its continuous Jacobi transform
F
#4a, B,
Let
(B,u)(>,)
-I
(-x) dx.
(x) p
Then
i)
(B,)(A)
is rapidly decreasing as
-
A
(R).
i) for any fixed real number r with 0 < r <
Ir(B,)(_1/2) f(Y), -(a,B)(y)> HO(
0
f(y),
where
HO()
Proof"
is given by
o
KA_1/2
p a B) (y)
(B,u) (x-)
(l-x2)y
which by
(2.7) can be
,_*
Thus, for any
PB’a)(-x)
[(-6)) + {+B+2)x]y’
satisfies the differential
-(X+u+B+l)y
written as
P.=)(-x)
k > 0
H(x) x sn x dx>
(2.2]).
i) From (2.3) it is obvious that
equation
X sinA dA
-x(+=++)
p.=)(-x)
.()
P’=)(-).
(3.2z)
A.I. ZAYED
684
(g,)()
E() k -I
[n()]
k
q(x) t *k P
dx
(B,)(},)
and from (2.15) we conclude that
ii)
B’)(-x)
-l
[n()] k -I
0(-)
B’)(-x)
P
for any
as
(x) dx
k > 0.
Let
Jr(Y)
jOlrP(’B)
>’’1/2
Jr (y) Ha,B’
Evidently,
Yk c(Jr (y))
<
(y)
(3.24)
for
l(l-y)-(l+y) c
sup
yc
(B,) (X-k) HO(>,) , sin , d>,.
r
0
(B,) (-1/2) H0(}’)
[q(,_1/2)]k p(,B)(y)
-}"1/2
sin
(3.25)
o,c
In fact
J(y)
is also in
first part of this lemma since
H,8 as can be seen from (3 25) and the
H0(},)
following Reimann sum for the integral
r m
Rm[Jr(Y)]
k
0(+-)
as
}, +-.
Jr(y)
,kr,
kr
kr kr
(B,) (---1/2)
p(,B)(y)
Ho(---)(---)sinT)"
"kr
As a function of y, each term in this sum belongs to
<f(y),Rm[Jr(y)] >
Construct the
is defined and in
f(Y)’Rm[Jr(Y)]>
addit!on
,B’
hence
to that we have
r m
"kr .>hk(r ,m)
k=l<fY)’ p(,B)
m
(3.26)
where
hk(r,m
(B,)(__kr
1/2)
kr
kr
H0{_)(__)sin.T..
Upon observing that the sum in (3.26) converges to the left hand side of (3.22)
as m
,
our lemma will be proved if we can show that
Y(,B as m-
.
Let
Rm[Jr(y)]
Jr(y)
in
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
h(X)
(B,)(_1/2) HO(),) >, sin (X).
_
Then
(l-Y)-Cl+y)C Lp
YP’c(Jr(Y)-Rm[Jr(Y)]) SUpy
{ IrP((’B)Cy)h()dJo -rk P(’B)(Y)h/(kr
. r,m)}
-1/2
r
hk
fl [rl(;_.)]P pC,B)(y)
k [- p!C,B),
K_
(l-Y)-a(l+y)c[k= [n(x’%)]p B)(Y)h(X)dX-.k=l[ ]p p(,B)(
_%kr hk
sup
y
sup
ycl
685
(l-y)-(l+y) c
--1/2
0
h(>,)
r
P
kr 1/2] p
r
d>,
Y
kr
Y)
r ,m)
(r
m
sup
yI
m
(1_y)_(1+y)C
[rl
(k-1)r
k
(-%)]PP-B) (y)h(;k)dx
"-- %
m
But since
of
{l-y)-(l+y)C[n{)]P
(y,),) in the region y
> O,
there exists
P_B)(y)h()
[-l,l] and ;
(3.27)
is a uniformly continuous function
[O,r], it follows that for any
m 0 large enough so that
m
Yp,c(Jr(Y) Rm[Jr(Y)]) < k=l2 ()
< r
for any m > m 0 and this proves our assertion.
Q.E.D.
4.
Characterization of the continuous Jacobi transform"
The main result of this section is theorem 4.1 which gives a characterization
of the continuous Jacobi transforms of members of the space
-I <B<
Theorem 4.1.
,B;
+B
0
A necessary and sufficient condition for a function F(),) to be
the continuous Jacobi transform of some generalized function in the space
according to definition
(3.6) is that F() is analytic in the half-plane
Re >, > 0 and bounded according to
686
A.I. ZAYED
IF(,)I <_ Q(I,I),
Q(II)
where
Proof"
is a polynomial in
Was proved in proposition 3.1.
Necessity"
m
and choose two positive integers
than the max
Re X > 0
and
n
Sufficiency"
such that
2m-n > q
deg Q
Let q
with
n
greater
{-,2-min(,B)}.
Since
c x1/2-n
Iv’-[n(>,-1/2)+l] -m F(>,-)I <_ c x-2m+q+1/2
for some positive constant
it follows that
C and sufficiently large X,
[n(X-1/2)+l] m
d;k <
b > O,
for any
and since the integrand is continuous and bounded on
V F(X-)
[O,b],
[q (:X-)+l ] m
L (O,oo)..
Therefore,by (2.19) and (2.20)
F(X-1/2)
[rl(X-)+]] m
:1/2
fl-lg(x) p(C,B)
where
g(x)
[n(’-1/2)+]]-mF(X-)
4
0
Hence,
sup
I(]-x)Cg(x)l <_
4
_
X dX
(-x) HO(X)
PX-(B’o)sin
I I[n(X-)+I]-mF(X-)
0
xel
(x)dx
where
sup
xI
I(]-x)
},
(X)
HO(X)X
dX
(4.1)
pB ) (-x)l.
But since
i[n(X-)+I]-mF(X-)(X)Ho(X)X
for some suitable constant
C,
<_ C
x-2m+qxmax(’B)X-’B+l
<C -n-min(,B)+l
it follows from our choice of
n
that the integral
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
in (4.1) is finite and hence
sup
l(l-x)g(x)l
Therefore,
<
xcl
for any
which implies that
0 < c <
(l-x)9(x)
(l+x) c
c
687
L I(-I,I)
by (3.2).
g(x)
Now set
(L * +I )m g(x)
f(x)
*
since L * is a continuous linear operator on
H,B
which is also in
H*
see (3.3)and (3.4).
-inally
1/2 f (x),
P_B)(x)
P_B) (x)>
1/2[n(-)+l]m<g(x) ,P_B)(x)>
1/2 (L*+I )mg (x),
<(x), (L+I }mP(’B)(x)>,
_1/2
[n(X-1/2)+l]m[n(X-1/2)+l]-mF(X-i)
F(-1/2).
Q.E.D.
5.
Inversion of the Jacobi transform-
The bulk of this section is devoted to deriving an inversion formula for the
Jacobi transforms of generalized functions.
generalizes the classical one given in
This inversion formula, of course,
[ 5].
Since the inversion formula for the continuous Jacobi transform given by
(2.19) and (2.20) are only valid for
+ B
O,
from now on we shall always
assume that this is indeed the case.
Lemma 5.1.
Let (y)
r
2
in
H, B
as r
o
.
I)(I) and
p(,B) (y)
-.x-
(B’)(X)
(X-1/2)
be given as in
He(X) X sin X dX
(3.21). Then
(y)
A.
688
Cr(y)
Clearly,
lira
2
0
Cr(y)
.
ZAYED
P-’(,) (y) (,) (X-)H O(X)Xsin
(5.2)
dX.
exists in view of lemma 3.2 part (i).
r/(R)
Let
(X.)
(x)
(’B)(n)
and
(l-x)(l+x) B
be its n th Jacobi coefficient as given by (2.17). Hence,
n=OZ
(x)
I
(n) P
(x)
(5.3)
n
where the series converges absolutely and uniformly on
P(1).
since
By combining (3.21) and (5.3), we obtain
which, in virtue of
_
(2.8) and (2.]]), yields
(B,)(_)
2
Z (_])n
hn
n=O
(,B)(n)B,
_
where the series converges absolutely and uniformly in
(’B) (n)
subset of [0,(R)) since
}n= 0
(n)
(5.4)
on any compact
is a rapidly decreasing sequence.
By substituting (5.4) into (5.2) and interchanging the integration and the
summation, one obtains
4n=OZ (-l)nhn
’rOY)
i) If
where
-I<B<
h n and
(,B)(n)
I
r
p,B)(y)B,)_ (n)H O(x)x
0
(5.5)
-,<=<I ), we use (2.12) to obtain
__,
-"
(,) (n)
n=O
n P_
(n) P
(y)
hn= O()
and
(5.6)
are given by (2.10) and (2.|1) respective]y. The serles in (5.6)
converges absolutely and uniformly on any compact subset of
fact that
sin,x dx
.(=:I) (n)
^
O(
n2_
because of (2.15) and the
as n
By substituting (5.6) into (5.5) and interchanging the integration and the
GENERALIZED INVERSION FORMULA FOR THE CONTINUOUS JACOBI TRANSFORM
689
summation, we obtian
r(y)
(-l)n
4
where
Ir(m,n)
0
(’B)(n)Pm( ’B)(y)I r (m,n)
h n hm
n ,m=O
r
f _B)(m)B)(n)Ho(},)_
sin; d
But Since
(-l) n
Iri_m Ir(m,n)
we deduce from
(y)
(c.f.[3])
hn6n,m
4
(5.7)
(5.7) and (5.3) that
:n
1__
=ohn
(e,B)(n)p :,B)(y)
n=O-n (,B)(n) P e,B)(y)
(l_y):(l+y)B Z
*(Y)
ii) If -I
-1/2
< e <
<
B
), we can, with some easy calculations,
<
use (2.19) and (2.20) with
f(x)
Pn(’B)(x)
to reduce (5.5) to
(5.8)
Interchanging the summation and the limit is possible since the series converge
uniformly on any compact subset of
To show that
Yk,c(r-)
0 as
Cr(X)
(-I,I).
(x) in
r
H, B
as
r
+
we need to show that
for every nonnegative integer
First, taking a note of
k and
> c >
(3.23) we make the following observation.
From this, it easily follows that
ty Kr(Y,X)
Therefore, for any positive integer
tX
Kr(Y,X ).
k we have
max(-B,O).
690
A.I. ZAYED
LyCr(y)
-l(X) LykKr(Y,x)d x
2
I
2
But since
Lk@
P(1),
-l
2
-l(x)
Kr(Y,x)dx
(Lxk@(x))Kr(Y,x)dx.
we can apply (5.1) to it and conclude that
im t k
r/
YCr(Y)
L k (y)
Y
and consequently
yk( r-@)
/
r
0 as
/-.
Q.E.D.
Now’we are able to prove our main result.
Theorem 5.1
H*
Let f(x)
and
Then in the sense of convergence in
f(x)
F ( B) ()
P*(1)
(a,B)(;.)p
lira 4
n/(R)
Proof:
be its continuous Jacobi transform.
)(-X)Ho(>,)
;ksinpdl.
Let (x) c P(1) and
fn(X)
We need to show that
(’B)(},-1/2)P(B’)(-x)Ho(,) sin d
4
fn’> /<f’>
as n
(.9)
+-.
By changing the order of integration and using lemma 3.2 we obtain
<fn,
4
=2
(’B)(;E-)
(,-1/2)
HO() }, sin
o<f(u), p_B)(u)> ( B,)(k-1/2) HO(>,)
<f(u),
2
(u)
Inp(’B)
0-;-1
(B’) (,-1/2)
d
k sin k dk
HO(>,) X sin 7>, d,
,
GENERALIZED INI/ERSION FORMULA FOR THE CONTINUOUS ACOBI TRANSFORM
whc,-e
7(6’u)(,)
is given by
is permitted since
(3.21). Interchanging the order of the integration
(x) has a compact support in
continuous function on a clos Lmunded domain.
of
(5.10) as n
/
591
and the integrand is a
Taking the limit of both sides
yields
by 1emma 5.1.
Q.E.D.
An immediate consequence of theorem 5.1 is the following weak version of a
uniqueness theorem:
Corollary 5.1.
Let f(x)
g(x)
*
and let F (o,e) (X)
be their continuous Jacobl transforms respectively.
in
Re
> O,
Corollary 5.2.
f(x)
n
then in the sense of equality in
Let f(x) be a regular member of
L -1 ,l)
f(x)
for almost all
4
If F
P*{I),
f
and G {’ ) {X)
(’B)(;)
G
(’6))()
g.
o,B such that
then
0F(c=’l)) (’k-)P(l;),k--t
(-x) HO(X) X sin X d,k
x ((-I,1).
Acknowledgement:
The author wishes to thank professor L.Debnath
for bringing to his attention the results of references [I] and [3].
69
AoIo oZA
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