Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 345-354
345
THE SECOND CONTINUOUS JACOBI TRANSFORM
E. Y. DEEBA
Department of Applied Mathematical Sciences
University of Houston-Downtown
Houston, Texas 77002
and
E. L. KOH
Department of Mathematics and Statistics
University of Regina
Regina, Canada $4S OA2
(Received February 14, 1984)
ABSTRACT:
This paper continues the work started in
[I];
a second
continuous Jacobi transform is defined for suitable functions f(x).
Properties of the transform are studied.
In particular, the first
continuous Jacobi transform in [i] and the second continuous Jacobi
transform are shown to be inverse to each other.
an extension of
The paper concludes with
Campbell’s sampling theorem [2].
KEY WORDS AD PHRASES:
Couinuous Jocobi traJform, inverse Jacobi
trs form, discrete Jacobi trans form, sampli,g theorem.
1980 ,THEMATICS SUBJECT CLASSIFICATION CODES.
I.
44A15, 33A30, 94A65.
INTRODUCTION.
In this paper, the second continuous Jacobi transform,
^f(a ,B),
of a
function f(x) is developed along similar lines of Butzer, Stens, and
Wehrens [3].
Debnath [4
The results generalize the work in [3] as well as the work of
on the discrete Jacobi transform.
Basic properties of
^f(a,B)
will be derived including an inversion formula tersely given by
(^f(a,)(.))^(a,)=f(.).
The results are then applied to an extended form of
Campbell’s sampling theorem [2].
The paper is divided as follows.
Section two includes basic notations
346
E. Y DEEBA AND E. L. KOH
Section three
and results obtained in [I] that will be used in the sequel.
is devoted to the study of the second continuous Jacobi transform.
In this
section, the first and the second continuous Jacobl transforms are shown to
be inverse to each other.
Section four is devoted to a sampling theorem
based on Jacobi transforms and an estimate of a truncation error.
PRELIMINARIES.
2.
In this section we recall all the necessary background material on
Jacobi functions and the first continuous Jacobi transform as studied in [i].
For the sake of completeness, we repeat some of the basic notions of hypergeometric functions.
For each a, b, c real numbers with c0,-i,-2
the hypergeometric
function is given by
F(a,b;c;z)=k=Ol
(a) k(b)
k
k
(C)kkW.
The above series converges absolutely and uniformly on each compact subset
of (-i,I).
Also,
lim F(a,b;c;z)--F(a,b;c;l)-zl
r(c) r(c-a-b)
F-(c-a)-F(c-b)
where the ganna function is always assumed to be a well-defined function of
its argument.
The Jacobi function is defined by
p a,) (x)-where a, 8>-1.
I’(+a+l
F(a+l)F(+l) F(-k, l+a+B+l
c+l;)
The following are some of the properties of
derived in [i] and will be used in the sequel.
xe (-1,1]
pa,B)(x)
We refer the reader to [I]
for proofs.
[Lemma 2.2, [1], page 148].
LEMMA 2.1.
i_>
a++l
2
(i)
For any x(-1,1) and
we have for
(x)
-l<_<0,
(ii) g>O,
(+a+l)
<r-f(a+l)
r(1) +M(,n,) log( _)
2
r(+a+)
) kiP a’a)Cx)[-< rc+)r(+)
+M’(k,a,B)log(-)
where M(,a,) and M’(,a,B) are constants depending on %, a, and
We will denote the weighted
w(x)--(l-x)a(l+x)
by
LP(-I
W
1).
LP(-I,I)
The norm on
B.
(p>l) space with welgbt
LP(-I
W
I) is given by
34?
SECOND CONTINUOUS JACOBI TRANSFORM
I
1
i
Eli p=(2+B+I
_lW(X)I f(x) IPdx)
l/p
It was shown in [4] that
(I.emma 2.4, [i], p.150).
LEMMA 2.2.
and for all
for all p>l
such that p+l>O and -I /p< 8< I /p
B
and
P’B)(x)ELwP(-I,I)
Jacobi polynomials with
%=nP, P
is the set of all positive
integers,
satisfy the following orthogonality relation
I
I
w(x) e(U,6) (x) e(a,6) (x) dx=
m
n
u++l
2
-i
f
where
I
n#m
0
n
n=m
r(n+a+l) r(n+B+l)
n!(2n+a+B+l) r(n++B+l)
n
However, Jacobi functions do not satisfy such an orthogonality relation.
Instead, the following result holds:
(;.emma 2.5, [i], p.150).
LE4A 2.3.
than
a+B+l
-v
2
and X-(u+a++l).
+I #-Iw(x)P
2
a,g)
r_(+a+l) r(++l)
(X-) (X++B+I)
Let
and u both be greater
For u, Be(-,-), we have
(x)
(-x) dx
sin
r(,+l)r(+a+B+l)
sin.
r(+l)r(v+a++l)
We also recall the definition of the discrete Jacobi transform as tudled
by Debnath in [4].
(x) f(x)dx
and under appropriate conditions
n=O
(2.2)
n
geL2w(-1,1)
Now, for any f,
and for appropriate
(2.3)
iJ-iw(x) f(x)g(x)dx n--O 6 -l(a B) (n)g^(,B) (n).
Lemma
together
P(na’B)(-x)=(-l)nP(n8’a) (x)
1
2
n
ad+l
From (2.3) and
with the identity
2.3
we ob rain
n
(n)=
r(x+a+l) r(n.+_+_l)sin%
(-n) (k+n+a++l) n r (+a+B+I)
%@n
(2.4)
F (nd<x+l) r (n+B+l)
2n+cx+B+l) n r (n d<x+B+l)
=n
E. Y. DEEBA AND E. L. KOH
348
pC,B) (x)
The following estimates on
will be used.
for large
For X,
(Lemma 2.6, [I], page 152).
LEMMA 2.4.
++I
>-------,
>-q and
-1/2<<1/2, we have
for each [a,b](-l,l), there holds for all xc[a,b]
(i)
IP
(il)
’{)
liP
(x)I=O(X -)
)(x)ll I
(iii) for each
X -*o.
as
(x)
2
o(x
-h)
xo.
as
,oo), there exists a constant M>O such
c,d]c_[- ++I
2
that for all X, v[c,d]
I1P :’)<x)-P <’(3)<x)il
v
2 <MI,-vl
The continuous Jacobi transform of the first kind is defined as
in
[1] by
Z)
(c
(i
for every
I
1
+i
2
fL2w(-1,1)
with
1
--1
w(x)P X
(2.5)
(x) f (x)dx
>-1/2 and -1/2<8<1/2.
We recall that if =8-0, we
X-nEP,
3] and if
obtain the results of Butzer, Stens and Wehrens
we obtain
the results of Debnath [4].
Again, it was shown in [i] that for any
If (c’ ) (x)l--o(x-)
as
fEL.2.(-I,I)
we have
(2.6)
X -,’.
and
(’)(’- a+l+l
+) nLP(IR+)
2 )Co(IR
(2.7)
p>2.
The following Lemma will be essential for our work.
LEMMA 2.5.
IR+=[0, o)
(Lemma 3.2,
such that X
G(x)-0
belongs to
++ F(X)LI(IR+).
Let F(x) be defined on
Then the function
F(X)P (’) .(-x) H() XsinnXdX
X +B+i
2
C(-I,I)nL__(-I,I)
H(X)
1], page 155).
where
F2tX" + +B+I.
2
r(x+ a-B+l
B-+I.
From now on we will assume throughout the paper that
and satfsfy
SECOND CONTINUOUS JACOBI TRANSFORM
349
(2.8)
If
3
and 8 satisfy (2.8), then it was shown in [I] that
_(a, B) (x)=4
ek
0
P a,) (’-) (,c’O (-X)Ho() sSnn)’d
(2.9)
PX-
where
r2(+)
H0(R) =r(+a+)r(++)’
x(-,], kP.
The following is also true.
LEMA 2.6. (Theorem 3.1, [I], page 158).
that
X(’B)(-)LI(IR+).
f(x)
-
(a,)
Let
fL2(-l,l)
be such
Then for almost every xE(-1,1)
(2.1:0)
(.j)F_ (_X)Ho()slnwd.
Moreover, if f(x) is continuous then (2.D)holds everywhere on (-I,I).
SECOND CONTINUOUS JACOBI TRANSFORM
3.
In this section we define the second continuous Jacobi transform
^f(a,) of a function f with a and 8 satisfying
^fCa, S) and f^(a,) are inverse of each other.
For each f defined on IR
(x)
+
(2.8). We also show that
we associate the integral
ffi4I(x)v,a)(_x)
r(),+,).
0
proosltlon 3.1.
e
ve
+
For each f defied on R Ith f and
(x)C(-,)nL2(-,).
thgh the
proo
of Proposition
.
alg the sa lines as Lena 2.5, we preset the proof with the proper
modificati for the sake of selfontaient.
PROOF of PrOposition 3.1.
We first show that l(x) is well defined.
rcx+)
that, for large l, r(l+8+) behaves llke
(see [i]).
T>O, we have by Lemma 2.4 (il) and the hypothesis that
1-8
l(x)l-<l
4
ITof(X)P_Ba)
(-x)
r(x+)
r(x++) Xsinxdxl
(,) (-x) r(+)
+l 4, f(),)’X_.t
r(/) slnXdl
-<Cl+C2 T[-+lfcx)ldX<
Observe
Thus for any fixed
350
E. Y. DEEBA AND E. L. KOH
where C
1
and C
2
are some positive constants.
We show that l(x)C(-l,l).
x+/-61c(-l,l).
Hence I is well-defined.
For any xc(-l,l), there exists a
By lemma 2.4 (i) and for all
lyl<l,
such that
we have
(-x)
(-x-y)
f
61>0
O
f(l+8)
0
by hypothesis and C is some positive constant.
Thus for 4
0
sufficiently
large
I
r(+)
(,a)(_x_y)_pa),
c be made sufficiently all.
Thus for >0 given, there Ists a
0
sufficiently large such that the above difference is less than /2. Fix
By t. oi.ity
o
P)-_ (-).
.,
h.e
o
>0 tt t.,. i,t,.
0"
2>0
such tt
-(,) (-x-y)p) (-x)[< whenever lyl<6 2.
[X
c=2 0&%[f()ldX and 6=min(gl,2). With this choice.
oose
0
[I(x+y)-I(x)[
whenever
[y[<.
We finally show that
Therefore I(x)C(-1,1).
I(x)L2w(-1,1).
and by Bardy-Littleed inequality (see [1], page 148) d La 2.4
ue have
llx(x)2a x-+lf(X)ld( -1 (X-x)
by hypothesis.
Bence
Z+x)IP_
I(x)cC(-1,1)L2w(-I,i).
(-x)12dx)
This completes the proof of
Proposition 3.1.
Ne sll call I(x) the second continuous Jacobi transfo and ee
will denote this by
f(a,).
f(a’) (x)=4
Thus, ee define
x)
(-x)
r(+)
SECOND CONTINUOUS JACOBI TRANSFORM
^f(’8)eC(-l,l)OL2w(-l,l) with -<u, 8<
From Proposition 3.1 we deduce that
and
351
In the next theorem, we will prove an inversion formula for
u+=O.
^f(u,8)
the transform
The inversion formula
under appropriate choice of f.
proof is analogous to the proof given in [3].
We will employ the Fourier
cosine transform of a function f which is given by
If()costd..
Fc(f)(t)=
feLl(IR +)
Assume that
THEOREM 3.1.
is such that
A-8+1/2f(A)eLl(IR +)
and that
Fc(f)(t)--O
for all -<t<
Then we have for almost all heiR
FCI+1/2)
1/2 (++1/2)
Moreover, if
fC(IR+),
I
1
.
+
w(x)^f (,8) (x)
-1
P
) (x)dx=f ().
then (3.1) holds for all AelR
(3.}
+.
By employing Fublni’s theorem together with Lemma 2.3
PROOF.
e+=O), we obtain
(with
1/2
r (++1/2)
2r(+)
r (++:)
F(+1/2)
=4 F
(%4<x+1/2)
w(x){4
f(t)P
o
-1
tf(t)sinwt
0
I
(8
t-
)(-x) r r(t+1/2)
(t+S+%)
r (t+1/2)
r(t+8+1/2)
{I
r(t+%)
r(t+s+1/2)
t f (t)slnt
I
w(x)
_I
P_
t sintdt
Pa_) (x)
dx
(x) -(8,e)
vt_1/2 (-x)dx}dt
r(,+a+%)
r(t+)
w(12-t2)
sin(t-1/2) -sin (l-l}) }dt
tf(t)sint{ cst-csWl} dtQ(l).
=4
w(12-t2)
0
Set
qt
cost-cos
2 -t 2
(
We claim that
Q()gLI(IR+).
Indeed, an application of Fubini’s theorem
together with the assumption that
fgLI(IR +)
implies that
0
Now, by another application of Fublni’s theorem, we obtain
352
E. Y. DEEBA AND E. L. KOH
FC[Q] (s)--
(k)coslsd
0
(t) slnt(
=4
=4
tf(t)sintqt(%)dt)cos%sd%
f (t) sint
(k) cos%sdk)dt
Fc(qt) (s)dt.
-
We employ the result of [3] and proceed analogously and determine that
Fc(qt)(s)=
Thus
FC( Q)(s)=2
sin([-s)t,
O-<s<1, t>O.
(t)sint sin(-s)tdt
FC (Q) (s)=Fc(f) (s)-Fc(f) (2-s)= Fc(f) (s)
Moreover,
Fc(Q)(s)=0
for
-<s<
.
The assumption
O-<s-<.
Fc(f)(s)--O
for w<s<
together with the uniqueness of the Fourler-cosine transform implies that
Q(%)=f(%) for almost every
-F(+)
% IR
+"
Thus
_lW(X) ^f
(’8) (x)P
)
(x) dx.
The continuity of f will imply that (3.1) holds for all %IR
+
From Theorem 3.1 and Lemma 2.6 we deduce that
(f(a,S) (.))^(a,8) <z)=2
r(%qx+)
f(%).
r(z+)
(3 2)
and
r(%+)
(r(++5) f^(,8)(.))(a,8)(x)__f(x).
Equations (3.2) and (3.3) reduce to the formulas obtained in
4.
(3 3)
[3] whenever e=8=O.
A SAMPLING THEOREM
In this section we give a proof of a sampling Theorem of Campbell [2] by
employing results on the continuous Jacobi transform of the first and second
kinds.
Moreover, we will obtain an
error estimate
for a function
f that is band-llmlted in the sense of the Fourler-coslne transform.
SECOND CONTINUOUS JACOBI TRANSFORM
If FC(IR
THEOREM 4.1.
i
F(>,) t
+)
353
is given by
IB)
w(x) f(x)p (c’ (x) dx
I),-%
(4.i)
-1
p
for some
>0
fCL(-l,l)
and
then one obtains for all heir
(2n+l) f(n+l) f(A+a+%)
F()=
n=O
-(n+1/2)
)
F(n,+P
n (2n+l) r(n+l)
n=O r(n+c+l) r(n+lB+l)
o
nn
r(v+a+) r(n++l)sin(A-)
(
212- (n+) 2)n
(X22-(n+)2) r(n+a+l)
n=O
<2n+l) F(n+l) r(,V+a+)s.in(Xp-(n+))
2) 7 (n++J) r (1+)
(
.
212-(n+)
for s>q for some
n,
F(n.+)
v
>0.
Let F satisfy hypotheses of Theorem 4.1 for some fixed
F(%)-(Sn,F) ()
<C(%+I
;=En(f)
where
En (f)=InfP
EP f-Pn2
is the set of all algebraic polynomials of degree n.
Denote by S f the n-th partial sum of
f(x)~
We give
In particular, we show that
Then there exists a constant C>O such that
PROOF.
and
F denote the n-th partial sum of the series in (4.2).
THEOREM 4.2.
Pn
F(-)
FLI(IR+), %-B+FLI(IR +)
below an error estimate for approximating F.
>0.
(a,) (n)
Theorem 3.1 when applied to the above situation yields
the series representation (4.2) for
Let S
sinr(),l-t)
n=O
REMARK 4.1.
Fc(F)(s)=0
(4.2)
(a,) (n)(a,) (n)
[ (-l)n(2n+l) r(n+l) r<x++s)
F(),)=
sin(-(n+))
F(n4+l) F(%W+)
(-i)
(2n+l) r(n+l)
L
n=O r(nd<x+l) r(n+3+l)
or
that
From (4.1), (2.3) and (2.4), we obtain (with +--0)
PROOF.
F(,)--
22
,n
2
+
I -l(x,)(n)p(Ct,)(x)
n
n
n= 0
354
E. Y. DEEBA AND E. L. KOH
2
where 6
!) (2n.+l
-1= (n
r (n+a+l) F (n++l)
n
Thus
(S n
(2k+l r (k+l) r(x++)
F) (X)
k=O (
22 -(k+) 2
F(k++l) F(+)
-
By Cauchy-Schwartz inequality and Lemma 2.4, we have
F(1)-(S
n,
<_%[i
F)(I)
w(x)
-i
<
f-S
n
ii
12=1% -iw (x)(f(x)
(S n f) (x))P
If(x)-(Sn f)(x)l 2 dx’
2< 2.f)lp
fI122 p(a,B) 2-En
llJ-1/2
LI
1
a,) (x)dxl 2
w(x) P (a’s) (x) 2 dx
’v-’
-1
2"
Thus
F(1)-(S
F)(1)
I-<CEn(f)(I+])-1/2
for some constant C.
The estimate obtained in Theorem 4.2 shows that the
error becomes smaller for large
ACKNOWLEDGEMENT.
The work of the first author was partially supported
by a grant from the University of Houston-Downtown while the work of the
second author was partially supported by NSERC of Canada under Grant A-7184.
References
I.
2.
3.
4.
5.
DEEBA, E.Y. and KOH, E.L. The Continuous Jacobi Transform,
Internat. J. Math. & Math. Sci., Vol. 6, No. 1 (1983), 145-160.
CAMPBELL, L.L. A Comparison of the Sampling Theorems of Kramer and
Whittaker, J. Soc. Induct. Appl. Math. 12 (1964), 117-130.
BUTZER, P.L., STENS, R.L. and WEHRENS, M. Continuous Legendre Transform, Internat. J. Math. & Math. Sci., Vol. 3, No. I (1980),
47-67.
DEBNATH, L. On Jacobi Transform, Bull. Cal. Math. Soc., Vol. 55
(1963) 113-120.
ERDELYI, A. MAGNUS, W., OBERHETTINGER F. and TRICOMI, F.G. Higher
Transcendental Functions, Vol. I, McGraw Hill, New York 1953.