57
Internat. J. Math. & Math. Sci.
(1985) 57-68
Vol. 8 No.
THE GENERAL CHAIN TRANSFORM AND
SELF-RECIPROCAL FUNCTIONS
CYRIL NASIM
Department of Mathematics and Statistics
The University of Calgary
Calgary, Alberta T2N IN4 Canada
(Received December 11, 1984)
ABSTRACT.
A theory of a generalized form of the chain transforms of order n is
developed, and various properties of these are established including the Parseval
relation.
Most known cases of the standard theory are derived as special cases.
Also a theory of self-reciprocal functions is given, based on these general chain
transforms; and relations among various classes of self-reciprocal functions are
established.
KEY WORDS AND PHRASES.
Fourier transform, ,hain trasforms, f’,eIZin transforr.Ts,
Parseval theorem, kernel functions, L2-space, self-reciprocal functions, gamma ]’unction, Riemann-Zeta function, Bessel functions, Whittaker functions.
1980 ,Ar{.L’,’TTCS SUBJECT CLASSIFICATION CODE. 44A05
i.
INTRODUCTION.
The standard and the simplest definition of a function g to be the integral
f
transforms of
with respect to k is that
f(t)k(xt)dt
g(x)
where k is called the kernal,
f(x)
I]
[1,VIII].
If further
g(t)k(xt)dt,
and g are pair of k-transforms.
f
then we say that
(1.2)
Under less stringent conditions
the equations (I.i) and (1.2) can be replaced by
p(t)dt
I
where
f(t)dt
k l(x)
I
I
t-l f(t)kl (Xt)t
(I .3)
t-lg(t)kl(Xt)d/
(1.4)
x
k(t)dt
The above reciprocity formulae define the pair
transforms.
f and g, genera|ly known as Watson
lany generalizations of this reciprocal relation have been given, and
C. NASIM
58
one of the directions was pointed out by Fox, [2], who introduced the idea of chain
He showed that for chains of the form
transforms.
I
g2(x)
gl(U)kl(UX)du
gn (x)
gn-1 (U)kn-1 (ux)du
gn(U)kn (ux)du
gl (x)
0
2, i.e., the Watson
there exists a theory similar to the standard theory for n
Also, Duggal, [3], considered the pair of equations.
transformations.
!
v()f(x)dx
z
(.5)
k()ff(z)
u
1
0
using convolution of the functions
f
If V
and 9.
v are taken
5
f
and characterizing the relatonship between
and
to be the Heavsde step functions, then
(1.5) and (i.6)
reduce to the equations sm[lar to the equations (1.1) and (1.2) above, thus extend-
ng
the
Watson’s
definition.
In ths paper e shall combine th- above mentioned to extensions of the usuai
e
atson integral transformations.
shall
gve
a generalized form of the chain trans-
forms of order n and develop properties praliel to hose of the standard theory.
Nost of the ell-kno results are deduced as special cases.
s
self-reciprocal functions
again many standard results are deduced as special cases.
re
gven
Further a theory of
developed based on these generai chain transforms; and
In the end various exapies
to llustrate the general nature of the main theory developed and its
special cases.
2.
PRELININARY SLTS.
Definition 2.1
[f();]
s
1
Let
n
<
Then define
f()z-
y()
+ g
integral exists
g2(0,).
f
<
.
F()
s
the mean-square sense.
(.)
f();
+ g),
said to be the Mell[n transform of
If further P()
I
g2(_
i,
the
then
(2.2)
F(s)-Sds
sense; we say that
the integral existing in the mean-square
1
<
[l,I].
<
i,
ellin transform of F(s), s
The Parseval theorem,
If
f
and g
is the inverse
=+
[l,III].
L2(O, )
(zt)(t)d
f(x)
,
respectively, then
and have Mellin transforms F(s) and G(s)
-g
’ a’-s’-d
(" )
THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS
A function
Definition 2.2.
k if
f
is its own
59
is said to be self-reciprocal with respect to the kernel
k-transform, satisfying the identity
f(t)k(xt)dt,
f(x)
0
We then say that
or its equivalent.
f
Rk
(2.4)
Although most of the results hold for functions involved under less stringent conditions, but we shall mainly work in the Hilbert space L 2, for convenience and elegance
of the theory of integral transforms in this setting.
CHAIN TRANSFOrmS.
3.
We consider the system of n integral equations
ki(ux)/i+l (x)dx
where i
vi(ux)fi(x)dx
n and f(x)
1,2
f().
For simplicity we shall assume that no two
are equal, although it is not essential.
We shall refer to the functions k. and v.
Now, we give one of our major results.
as kernels.
Theorem 3.1.
If
(i)
(ii)
(iii)
7,’.i,vi,fi
and
ki(ux)fi+l (x)dx
}(x 71 f() and
n
n V/(s) E
S(s)
i=
uo
then
and
fn+ (x)d
P (x)
uhere
Vg()
and
fn+l
2-
Kg()
1
+ g,
0
1/2-i,
Ki(s)
L2(0,=), i
all
(ux)fi(x)dx,
i
n.
1,2
n where
(3.1)
L
1
(-
, + i(R)),
1
(3.2)
x-lP (u) (x)dx
1
P() x
(. 3)
d
denote the Neiln transforms of
<
1,2
.
vg(z)
PROOF.
and
kg(z)
respectively on
Due to the hypothesis () and the fact that
g(0,) henever
the ntegrals in (3.1) exist and are n fact
absoluteIy convergent.
the 2-functons n () have MellOn transforms h[ch
belong to
g,
+ g) due to definition (2.1) above.
fi
fi
(0,),
Also,1
2(1
No due to the Parsevai theorem for MellOn transforms of
boh sides of (3.1), we have
Ki(s)Fi+ 1(s)
on s
1
we obtain
+ i,
<
(s)Fi(s
=, i
1,2
a.e.
n.
2-functtons,
applied to
(3.5)
On successive elimination and using (3.2),
C. NASIM
60
y (s)
P()Yl(S)
n+l
Next we have
M-l
,f-+l(x)
F (8)
x],
n+l
therefore
[%+{" P(S)fl (s)x-Sds’
1
fn+l (x)
i.
(3.6)
ioo
the integral converging in the mean-square sense, since
I
i
F I(8)
+ fro). 0r from (3.6), we have
i,
1
P(
+ iT)
is bounded and
L2(
u
F
=I
(x)
P(s)F (s)
u -s
the integration inside the integral sign is justified because of uniform convergence
due to hypotheses (i) and (iii).
1
1
+ i ) and hence
z,
L2(-
X
-
exists and
Now since P(s) is bounded, therefore P(s)
1
8
Pl (x) w-[LlL2(0,). Also
(s)
fl(x)
I[F l(l_s);
U
xl,
therefore due to the Parseval theorem applied to the right-hand side of (3.7), we
obtain
u
x- Ip (ux)
fn+l (x)dx
(x)dx
as desired.
DEFINITION 3.1.
The sequence of functions
{fi
is said to be the general chain
transform of order n, with respect to the kernels k
the system (3.1) along with (3.3) is satisfied
and vi, i
1,2,...,n, whenever
i
It is now an easy matter, to prove a
stronger result than that of Theorem i, which we give below without proof.
Let the conditions of (i) and (ii) of THEOREM i hold.
THEOREM 3.2.
Then
necessary and sufficient condition that (3.3) holds is that
v.(s)
n
P(s)
7
I-i
=1
.gi ()
Next we shall give some special cases of the result of THEOREM i.
I.
First let n
COROLLARY i.
If
i ()72 (X)C
and
V
(2)i (X)C
V (s)
P(s)
then
f2 (x)dx
where
Pl (X)
1
x- (ux)
[+i
i
(x)dx
P(s)_ s xl-s
ds.
(3.8)
THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS
Thus if k 1,v
and
function
If we further set
2"
fl
0
v
61
are known, we have an inversion formula to retrlve the
unknown
if 0<x< i,
(x)
i
if x> i
we obtain a familiar unsymmetric transformation of
COROLLARY 2. If
u
I fl
I(R)u kl )f2
u
(X)C
Watson’s type [1,VIII]; that
is
(x)o
and
1
P(s )K (s)
1
then
Pl (U)fl (X)C
f2 (X)OL
Next let n
2.
Then THEOREM 2, reduces to a known result [3],
that is:
If (i)
and
E L2(0,), i
1,2,
COROLLARY 3.
ki,fi
f3
and
then
x-lpl (ux)fl (x)dx
f3 (x)dx
where
Suppose, now,
1%+i P(s) xl-S ds
1
P (x)
J1/2_iil- Z
Pl(X)
p (x)
is differentiable, then from
(3.4), we have
p(s)x-Sds
p (X)
1/2-i
i.e.
p(x)
-i
M
[P(s); x].
Further let
k i(x)
i
x),
(i
1,2,...,n and 6 being the Dirac delta function.
Then the sequence (3.1), reduces
to
fi+1 (u)
v
i (um)fi
(x)cLz
and consequently we have, a non-integrated version of Theorem 1:
THEOREM 3.3.
If
(i)
(iii)
vi,fi
P(s)
and
H
i=I
fn+l
Vi(8),
L2(0’)’
i
1,2
n
C. NASIM
62
then
n+l (u)
p(x)
where
Letting
p(x)
o
--v
p()71
P(s)x ds
|
(l-x), and hence, its Mellin transform, P(s)
1 the above result,
then, defines the chain transforms introduced by Fox.
Note that all the above
mentioned results are specializations of our Theorem i, showing the general nature of
that result. Next we shall deduce a Perseval type relation for the general chain
transforms as defined by Definition 3. Let sequences
forms of order n with respect to the kernels k. and v..
{fi
and
{gi}
be chain trans-
Now consider
where
+
+
()
()
By virtue of THEOREM i and the consequence (3.3), we have
x-lf:+1(ux)1(x)dx
x-11(x)dx
t-l
t-lp (uxt)?1(t)dt
(t)dt
x
0
t
-lpl (t)l
(t)gn+l (ut)dt
Hence
x-l+*
t-l ()gn+l (ut)dt
(uz)g (x)d
establishing, formally, the Parseval relation.
(3.9)
Further on differentiating both sides,
formally, we have
71 (t)L+1 (ut)dt
fn+1 (ux)1 (x)
1, reduces the above to the usual arseval Theorem, [1,
It is now an east matter to develop an inversion theor for the general chain
transforms. For instance Sf we assume that
ettng n
P(s)P(I
)
i,
then from (3.3), due to the standard inversion formula, we have
1 (X),
x-lpl (zeX)fn+l (x)
Thus the unknown function
1
EO,
P(e
E0,
0
I
is retrieved.
-x
On the other hand if we assume that
d
being the Lagunerre-Polya class [4, VII], then due to the known inversion tech-
niques, [5]
1
p(o)
[fn+l (x)]
giving us the
fl (x)
function
I"
THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS
4.
63
SELF-RECIPROCAL FUNCTIONS.
is a sequence of general chain transforms of order n, with respe-t
{i
Suppose
to the kernels
v..z
kz. and
If we then set
L2(O,),
ki,vi,i
If (i)
THEOREM 4.1.
1,2
i
and
n-l,
then
in Theorem
I,
we have
1,2
(x)
f(
LI(
i,
n V.(s)
z
H K.(8)
i=i
P()
(iii)
n+l(X) l(X)’
1
0
d.
1
From the conclusion (4.3), t follows that
kernel
Pl(z)’
s
The particular case when n
t,.
Sbol[cally we say that
in the sense of definition 2.2.
fl
self-reciprocal with respect to
R
P
In this case, many useful
2 is of special interest.
properties of self-reciprocal functions are established, including a procedure for
generating self-reciprocal functions with respect to a given kernel.
Now let n
2.
Then,
COROLLARY i.
If (i)
(ii)
and
ki,v i
f
L2(0’)’
fi
f
(um)2(x)o!r
k
k 2 (ux) fl (x)dm
i
(um)l (x)ci
V2 ()2 (x)
P()
then
i
PROOF.
,
(4.5)
+ g)
R
Pl ()
where
c(
(
(4.4)
V
and
()
1,2,
_g
By the Parseval theore for
1
Nell[n
transforms, equation (4.4) and (4.5)
give respectively,
()F()
()F ()
()F (-)
i
a.e. on
tons.
+ i,
(4.)
(4.7)
() (),
<
<
,
invlng the Nell[n transfor of the respective func-
Next define functions
C()
and
((_
)
)
()
()
()
(),
C
(g-
, g+
+ i,
i)
.
C. NASIM
64
Then from (4.6) and (4.7),weobtain, a.e. on
I +
iT,
8
,
< T <
F2(8)
M(8)F2(I
8)
(4.8)
F I(8)
L(8)F2(I
8).
(4.9)
and
Also,
L(I- 8) M(8)
L()
P(8)
By the Parseval theorem for Mellin transform of
L2-functions (4.8)
and (4.9) give,
re spec t ively,
x-lm(um) f2
f2 (X)Clr
IU fl
I x-
(x)dx
gl (UX) f2 (x)dx
where
and
e (X)
[1/2+i 1(8) x
1
m l(x)
= [1/2+i
L(s)
1
1/2-i
The functions m
e
and
1-s
dx
3:1-8
ds
8
)1/2-i
1
S
are defined by the integrals which exist in the mean-square
sense, since both M(8) and L(8) are bounded on
1
8
+ i,
<
M(8)
< =, due to the
and
hypothesis (iii) of Corollary 1 above, and consequently
1
iL
1
1
+ i). Thus, combining the above results, we have,
i,
L(8)
both
8
L2(
Corollary 2.
fl
(i)
If
and
L2(0’)’
f2
then
fl
where
gl,ml
and
Pl
R
are defined above.
This result gives us a procedure for generating a new class of self-reciprocal
functions, given a self-reciprocal function and the kernels.
(ii) shows that
I
2
Note that the hypothesis
If all the functions involved are differentiable, we obtain
a simpler form of the Corollary 2 above, that is:
Corollary 3.
If
(i)
fl
and
f2
L2(0,)
(ii)
-2 (u)
m()f2 (x)o,
(iii)
fl (u)
e()f2 (x)o
(iv)
P(8)
and
L(1
L(s)
(i.e.
f2 Rm)’
THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS
65
then
For example, let
re(x)
Now
-2[si+J(4x1/2) + co+{y(4x’) K(4x1/2)}],
1
(I
s -,)(i
(s)
i
where
[
(z)
n
-z
s +,)
(8 +
-2-)
(8
R(z)
_> 0
1
i
)
the Riemann-Zeta function
i
Also
n=l
F(I
1
1
i
(2) 1-28(8 +
)cos (8 + )
l
l
l
+ -)sin
(8 1
M(8)
8
by making use of the identity
(8)
8-128F( I
98
8)sin
8).
(i
It is an easy matter to see that
M(8)(l- 8)
l,
hence m(m) is a Fourier-Watson kernel.
L(8)
(8
cosec
Further let
1
-),
then
1
2+
and
p(x)
I[p(s);
M
1/2)
i
2 J (4x
x]
< Re 8 <
Thus,
Corollary 4.
If (i)
(ii)
f
Rm,
g(x)=
m defined by (4.10),
[
2
where 8(x)
then
(X)
Rp,
where p(X)
1
2
+ x2
J(4).
In particular, let
f(x)
1/2 cos(2X).
X
f
One can show that
g(x)
I
2
x
m"
Then
(xt)f(t)d
X
-1/2
I cs(2$)2
i + x2t
dt
(I+7#) -2lx
e
[6;221(55)].
Now
p(xt)g(t)dt
2
t
e
-2t
J (4
x’t1/2)dt
3
(4.10)
C. NASIM
66
xC-2?x
[6;29(10)].
g (x),
Corollary 4.
as predicted, thus verifying the result of
different version of Theorem
I
4.1,
so that the function
Next we shall give a slightly
fl’
rather than the function
is involved in the conclusion.
Let the conditions (i) and (ii) of Theorem 4.1 hold.
THEOREM 4.2.
n V.(l
s)
z
U
-s)
i:i K/(i
Q(s)
then
where
o
f (x)
qi(x)
f
i
(-
x- ql (ux)f
= "e
Q(s)
1
_i
x
s
i
Again the special case when n
i,
i
i)
+
(x)Ix
i-
2,
Further if
(r
f
R
ql
ds
is interesting, since it gives us a procedure for
generating self-reciprocal functions.
COROLRY 5.
(i)
If
fi,vi
()
and k
1,2,
(z)
()f()
k ()f ()
e()f()
()
then
L (0,), i
i
(
)(
ru
)
q
.e.
f
(R
q
Now from the hypothesis ([[), by using the Parseval theore
or
Nellin transfos, the
to equations imply respectively,
KI(8)E2(8
VI(8)FI(8
K2(8)FI(1
a.e., where
8
V2(8)F2(8),
8)
1
+
< T <
T,
.
If we consider the functions L(8) and
F2(8)
M(s)F2(I
s)
El(S)
L(s)F2(I
s),
M(s),
defined earlier, then as before, we obtain
and
a.e.
The last two equations, then give, respectively,
and
0
f2(x)dx
fl(X)dz
where
I (X)
i
IIm(ux)f2(m)dzxx1/2+i
I
#_ioo
(ux)f2(x)dx
L(s)
S
xi-Sds
(4.11)
(4.12)
THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS
and
1
m 1(x)
=-i-4
+
M(.8)
xl-Sd8
s
1
1/2-i
67
Thus we, formally, have
COROLLARY 6.
(i)
If
fl
and
f2
6
L2(O,==),
(4.11) and (4.12) are satisfied,
L(8)
(iii) Q(8)
8)
L(I
8) M(I
(ii)
then
u
i x- q l(uZ)fl(z)
fl(x)/
where
I’
ml’ql
are as defined above.
It is not difficult to prove the above result rigorously, the functions
and
1
and M(8), L(s) and Q(8) are bounded on 8
<
T <
+ iT,
hence the
Parseval theorem can be applied to give us the desired result. A non-lntegrated fo
of the above result is as follows:
,
L2(0, )
COROLLARY 7.
then
J
fl(u)
The kernel
(i)
f2(u)
(ii)
fl(U)
If
0
I
fl
f2
m(ux)f2(x)dx
e(um)f2
q(uz)fl(x)c5:.
0
ql (u)
q(z)ce
and the kernels (z) and m(m) are defined similarly. Note that the above results
gives
us a procedure for generating self-reciprocal functions. For example, let
re(x)
x
J (x)
then its Mellin transform is given by
1
M(s)
2
s-
F(} +
1
+
)
3
+1
1
r(
[7; 326(1)]
Now, let
(xI2)1/2(++I)K+1/2 (_) (x)
(x)
K
being the usual Bessel function of third
2 8-
L(8)
r(1/2+ +Z-)
1
1
Therefore
L(8)
L(i Z 8) M(1
(s)
2-1/2
whence
q(x)
r(1/2
1
(gu
1
kind and
1
+ /Z-)
8)
+ +)
1
3
1/2 J
(x).
x
Thus, we have as a special case of Corollary 7.
COROLLARY 8.
1
(x)
0
(xt
If
f
H
7F(++1)
and
(._) (xt)f(t)dt,
its Mellin transform is
C. NASIM
68
then
Here the class
H
Hankel transforms of order
x v+ e -2
f(x)
and
H
f(x)
.
denotes the class of self-reciprocal functions with respect to the
To varify this result, let
Then, [6; 132(25)],
I(-"-")r(+z)r(
_I
g(x)
_1/2(P++I)
(:t)
K1/2(,_) (xt)f(t)dt
I
e
Is n ey mae o
d
cul ce
o
I
i
+ 1)x1/2 (v+- ex2 /4W_(++),(_) (x/)
b g(x)
obed,
be boe eul
CORO 9.
+
H
e
peieed, [6;84(15)].
e
A i-
i.e.
R
f
e_Xtf(t) dt
g(x)
then
Rc,
g(x)
where R
and R
denote the classes of self-reciprocal functions with respect to Fourier
sine and Fourier cosine
transforms, respectively.
In the operational notation, we have
that if
F [f]
f
and
L[f]
g,
then
F [g]
where
Fs,Fc
respectively.
f(x)
and L denote the Fourier sine, Fourier cosine and the Laplace transforms
+1/2 2
1"(41-1) x
23
such that
f(x)
-
For example, consider
e
-x 2 /
M3, (x2/2),
Re
>
i
4
R8’ [7 ;115(5)].
Now,
2-X 29-I
[7;215(13)]
as predicted.
where M
"’2/4
’
-3),
and W
-
(x2/O)
are Whittaker’s functions and then
g(x)
Re,
[7;61(7)],
REFERENCES
i.
2.
3.
4.
5.
E.C. Titchmarsh, Fourier Integrals, Clarendon Press, Oxford, 1948.
C. Fox, Chain transforms, J. London Math. Soc., 2_3 (1948), 229-235.
B.P. Duggal, On Fourier transform type operators, Period. Math. Hungary, 9 (1978),
55-61.
D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971.
C. Nasim, An inversion formula for a class of integral transforms, J. Math. Anal.
Appl. 52 (1975), 525-537.
6.
7.
A. Erdelyi, et al., Tables of Integral Transforms II, McGraw-Hill, New York, 1953.
A. Erdelyi, et al., Tables of Integral Transforms I, McGraw-Hill, New York, 1953.