International Journal of Analysis and Applications
ISSN 2291-8639
Volume 3, Number 2 (2013), 81-92
http://www.etamaths.com

THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS

OF ARBITRARY ORDER

B.B. WAPHARE∗ AND S.B. GUNJAL

Abstract. In this paper four self reciprocal integral transformations of Hankel

type are defined through

(Hi,α,βf)(y) = Fi(y) =
∫ ∞
0

pi(x)gi,α,β(xy)f(x)dx,H
−1
i,α,β

= Hi,α,β,

where i = 1, 2, 3, 4; (α−β) ≥ 0, p1(x) = x4α, g1,α,β(x) = x−(α−β)Jα−β(x), Jα−β(x)
being the Bessel function of the first kind of order (α−β), p2(x) = x4β, g2,α,β(x) =
(−1)α−βx2(α−β)g1,α,β(x), p3(x) = x−4α, g3,α,β(x) = x4αg1,α,β(x) and
p4(x) = x

−4β, g4,α,β(x) = (−1)α−βxg1,α,β(x). The simultaneous use of trans-
formations H1,α,β and H2,α,β (which are denoted by Hα,β) allows us to solve
many problems of Mathematical Physics involving the differential operator

∆α,β = D
2 + 4αx−1D, whereas the pair of transformations H3,α,β and H4,α,β

(which we express by Hα,β) permits us to tackle those problems containing its
adjoint operator ∆∗

α,β
= D2 − (4α)x−1D + 4αx−2, no matter what the real

value of α − β be. These transformations are also investigated in a space of
generalized functions according to the mixed Parseval equation∫ ∞

0
f(x)g(x)dx =

∫ ∞
0

(Hα,βf)(y)(Hα,βg)(y)dy,

which is now valid for all real α−β.

1. Introduction:

Following Zemanian [15, 17], it can be proved that the Hankel type transforma-
tion of order (α−β) ≥−1

2

(1.1) (hα,βf)(y) =

∫ ∞
0

(xy)α+βJα−β(xy)f(x)dx,

where Jα−β(x) denotes the Bessel type function of the first kind is an automorphism
on the space Hα,β of infinitely differentiable complex-valued functions φ(x), x ∈
(0,∞) such that

ρ
α,β
m,k(φ) = sup

0<x<∞

∣∣xm(x−1D)kx2β−1φ(x)∣∣
exists for each pair of non-negative integers m and k. Then the generalized Hankel
type transformation h′α,β is defined on the dual space H

′
α,β by means of the adjoint

2010 Mathematics Subject Classification. 46F12, 44A05.
Key words and phrases. Complementary Hankel type transformation, Parseval equation, gen-

eralized functions.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

81



82 WAPHARE AND GUNJAL

operator of hα,β that is,〈
h′α,βf, Φ

〉
=
〈
f,hα,βΦ

〉
, f ∈ H′α,β, Φ ∈ Hα,β.

We emphasize that the last expression appears as a generalization of a well-known
Parseval equation for the Hankel type transformation ([17, p. 127],[6]). Conse-
quently, the generalized transformation h′α,β is also an automorphism on the space

H′α,β provided that (α − β) ≥ −
1
2
. Zemanian [16], Koh [4] later on remove the

restriction (α−β) ≥−1
2

and assume that (α−β) be any fixed real number. For it,
a positive integer r is chosen such that α−β + r ≥−1

2
and then the Hankel type

transformation of arbitrary order α−β is given by

(hα,β,rφ)(y) = (−1)ry−r[hα,β,r(Nα,β,r−1 . . . . . .Nα,β,1Nα,βφ)](y),

where the operator

Nα,β = x
2αDx2β−1

generates an isomorphism from the space Hα,β onto Hα,β,1 and the mapping φ(x) →
xrφ(x) is an isomorphism between the spaces Hα,β and Hα,β,r as well. The main
idea of this method consists in leading any member φ(x) ∈ Hα,β by means of the
applications of the operator Nα,β,Nα,β,1, . . . . . . and Nα,β,r−1 one after another to
the space Hα,β,r where the inversion formula of the Hankel type transform has
already a sense since α−β + r ≥ 1

2
.

This procedure is employed to extend to an arbitrary order α −β. The following
variant of the Hankel type transformation (see [8]):

(1.2) (H1,α,βf)(y) = F1(y) =
∫ ∞
0

x4αg1,α,β(xy)f(x)dx.

where g1,α,β(x) = x
−(α−β)Jα−β(x). The transform (1.2) is called in the available

literature the Schwartz’s Hankel type transformation.
The main objective of this paper is to extend the transformation (1.2) to any real

value of α −β through a technique quite different from that used in the previous
works [4],[8],[16].

To attain a more symmetrical expression for our results, from now on we assume
that (α − β) ≥ 0. L(I) denotes the space of all functions f(x) that are Lebesgue
integrable on the real positive axis I = (0,∞). D(I) stands for the space of infinitely
differentiable functions whose supports are contained in I and its dual D′(I) is
the space of Schwartz distributions [11]. Finally, E(I) represents the space of all
infinitely differentiable functions on I and its dual E′(I) is the space of distributions
with compact supports [17, p. 36].

2. Classical results on Schwartz’s Hankel type transformations:

We establish the following in relation with the transformation (1.2).

Theorem 2.1. Let (α−β) ≥ 0. If y2αf(y) ∈ L(I), then∫ ∞
0

y4αg1,α,β(xy)F1(y)dy =
1

2
[f(x + 0) + f(x− 0)],

in a neighborhood of every point y = x where f(y) is of bounded variation.



THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS 83

Proof. The outcome of the theorem follows from the relation

(H1,α,βf)(y) = y2β−1
{
hα,β

(
x2αf(x)

)}
(y)

between (1.1) and (1.2) and from the inversion theorem for the Hankel type trans-
formation [13, p. 240]. �

Other conditions under which Theorem 2.1 holds were proposed by A.L. Schwartz
[12].
Note that the function g1,α,β(xy) satisfies the equation.

(2.1) (∆1,α,β,x + y
2)g1,α,β(xy) = 0.

where

∆1,α,β,x = ∆1,α,β = D
2 + (4α)x−1D, D ≡

d

dx
.

If f(x) and G1(y) are functions defined on I such that x
2αG1(y) ∈ L(I) and if

F1(y) = (H1,α,βf)(y) and g(x) = (H−11,α,βG1)(x) = (H1,α,βG1)(x), where (α−β) ≥
0, then we easily obtain the Parseval equation

(2.2)

∫ ∞
0

x4αf(x)g(x)dx =

∫ ∞
0

y4αF1(y)G1(y)dy.

In past years, Schwartz’s Hankel transformation has been investigated in certain
spaces of distributions, amongst other authors, by L.S. Dubey and J.N. Pandey [2],
W.Y.Lee [5], A. Schuitman [10] and G. Altenburg [1].
Further, we introduce the transformation

(2.3) (H2,α,βg)(y) = G2(y) =
∫ ∞
0

x4βg2,α,β(xy)g(x)dx.

where g2,α,β(x) = (−1)α−βx2(α−β)g1,α,β(x) fullfils the equation

(2.4) (∆2,α,β,x + y
2)g2,α,β(xy) = 0,

∆2,α,β denotes the differential operator

∆2,α,β,x = ∆2,α,β = D
2 + 4βx−1D.

observe that the multiplication x2(α−β) only implies the change of sign for the
parameter (α − β) in (2.1) to convert it into equation (2.4). On the other hand
transformations (1.2) and (2.3) are closely related as it is made evident by

(2.5) (H2,α,βg)(y) = (−1)α−βy2(α−β)
[
H1,α,β

(
x−2(α−β)g(x)

)]
(y).

Then from Theorem 2.1 and (2.5) it is immediately inferred the classical inversion
formula for the transformation H2,α,β:

Theorem 2.2. Let (α−β) ≥ 0. If y2βg(y) ∈ L(I) and G2(y) is defined as in (2.3),
then we have ∫ ∞

0

y4βg2,α,β(xy)G2(y)dy =
1

2
[g(x + 0) + g(x− 0)]



84 WAPHARE AND GUNJAL

in a neighborhood of every point y = x where g(y) is of bounded variation. For a pair
functions f(x) and G2(y) such that x

2βf(x) and y2βG2(y) ∈ L(I), the following
Parseval equation

(2.6)

∫ ∞
0

x4βf(x)g(x)dx =

∫ ∞
0

y4βF2(y)G2(y)dy

is valid, where F2(y) = (H2,α,βf)(y), g(x) = (H−12,α,βG2)(x) = (H2,α,βG2)(x) and
(α−β) ≥ 0.

According to the relation (2.5) the transforms H1,α,β and H2,α,β coincide seem-
ingly when and only when α − β = 0. However, it is worth to remark that there
are other values of α−β which make equal the transformations H1,α,β and H2,α,β.
Infact, from the relation [13, p. 16]

g1,−n(x) = g2,n, n = 0, 1, 2, 3, . . . . . . ,

we quickly deduce that
(H1,−nf)(y) = (H2,nf)(y),

in other words, the transformation H2,n of positive integer order might be used to
replace the transformation H1,α,β of negative integer index.
This fact and above considerations suggest to adopt the notation

(2.7) ∆α,β = ∆α,β,x = D
2 + (4α)x−1D = x−4αDx4αD, −∞ < (α−β) < ∞

and

bα−β(x) =

{
g1,α,β(x), if (α−β) ≥ 0
g2,−α,β(x), if (α−β) < 0

.

Now we assemble (1.2) and (2.3) in the unique expression

(2.8) (Hα,βf)(y) = F(y) =
∫ ∞
0

x4αbα−β(xy)f(x)dx.

Thus Theorem 2.1 and 2.2 can be enunciated together:

Theorem 2.3. Let (α−β) be an arbitrary real number. If the function f(y) is such
that x2αf(x) ∈ L(I), if f(y) is of bounded variation in a neighborhood of y = x ∈ I
and if F(y) is given by (2.8), then∫ ∞

0

y4αbα−β(xy)F(y)dy =
1

2
[f(x + 0) + f(x− 0)].

Following [7], this other variant of the Hankel type transformation

(2.9) (H3,α,βf)(y) = F3(y) =
∫ ∞
0

x−4αg3,α,β(xy)f(x)dx,

where g3,α,β(x) = x
4αg1,α,β turns out to be a solution of the equation

(2.10) (∆3,α,β,x + y
2)g3,α,β(xy) = 0,

∆3,α,β being the differential operator

∆3,α,β,x = ∆3,α,β = D
2 − (4α)x−1D + (4α)x−2.



THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS 85

The following inversion formula follows from [7]:

Theorem 2.4. Let (α − β) ≥ 0. If f(y) is a function defined on I such that
y2β−1f(y) ∈ L(I), then∫ ∞

0

y−4αg3,α,β(xy)F3(y)dy =
1

2
[f(x + 0) + f(x− 0)]

in a neighborhood of every point y = x where f(y) is of bounded variation.
Moreover, if f(x) and G3(y) are two functions defined over the positive real axis

such that x2β−1f(x) and y2β−1G3(y) ∈ L(I), we obtain the Parseval equation

(2.11)

∫ ∞
0

x−4αf(x)g(x)dx =

∫ ∞
0

y−4αF3(y)G3(y)dy.

where F3(y) = (H3,α,βf)(y), g(x) = (H−13,α,βG3)(x) = (H3,α,βG3)(x) and (α−β) ≥
0.

Finally, we introduce a fourth integral transformation by means of,

(2.12) (H4,α,βg)(y) = G4(y) =
∫ ∞
0

x−4βg4,α,β(xy)g(x)dx,

in those kernel appears the function

g4,α,β(x) = (−1)α−βxg1,α,β(x) = (−1)α−βx−2(α−β)g3,α,β(x),

(2.13) (∆4,α,β,x + y
2)g4,α,β(xy) = 0.

Here ∆4,α,β symbolizes the differential operator

∆4,α,β,x = ∆4,α,β = D
2 − (4β)x−1D + (4β)x−2.

From the relation

(2.14) (H4,α,βg)(y) = (−1)α−βy
[
H1,α,β

(
x−1g(x)

)]
(y)

between transformations (1.2) and (2.12), by virtue of Theorem 2.1, we can easily
state the inversion formula for the new transform:

Theorem 2.5. Let (α −β) ≥ 0. If g(y) is a function on I such that y−2βg(y) ∈
L(I), if g(y) is of bounded variation in a neighborhood of the point y = x ∈ I and
G4(y) is given by (2.12), then∫ ∞

0

y−4βg4,α,β(xy)G4(y)dy =
1

2
[g(x + 0) + g(x− 0)].

Another result, which we shall need, is the corresponding Parseval equation that
now takes the form

(2.15)

∫ ∞
0

x−4βf(x)g(x)dx =

∫ ∞
0

y−4βF4(y)G4(y)dy.

where F4(y) = (H4,α,βf)(y), g(x) = (H−14,α,βG4)(x) = (H4,α,βG4)(x) and (α−β) ≥
0,f(x) and G4(y) being a pair of functions such that x

−2βf(x) and y−2βG4(y) ∈



86 WAPHARE AND GUNJAL

L(I).
By pursuing Theorem 2.4 and 2.5, Parseval relations (2.11) and (2.15) and d-

ifferential operators ∆3,α,β and ∆4,α,β the unique apparent difference lies in the
change of the sign on the parameter (α−β). In the same way it is convenient to
point that the multiplication of g3,α,β(x) by (−1)α−βx−2(α−β) to get the function
g4,α,β(x) exactly implies the said change of sign. Furthermore, it is easily seen that

(H3,−ng)(y) = (H4,ng)(y),

since g3,−n(x) = g4,n(x), n = 0, 1, 2, 3, . . . . . . ., now, it is wholly justified to adopt
the following notation
(2.16)
∆∗α,β = ∆4,α,β = D

2 − (4α)x−1D + (4α)x−2 = Dx4αDx−4α, −∞ < (α−β) < ∞

and

b∗α−β(x) =

{
g3,α,β(x), if (α−β) ≥ 0
g4,−α,−β(x), if (α−β) ≤ 0

.

Thus, the transformation (2.9) and (2.12) can be rewritten in the unique expression

(2.17) (H∗α,βg)(y) = G
∗(y) =

∫ ∞
0

x−4αb∗α−β(xy)g(x)dx.

Now, we summarize Theorems 2.4 and 2.5.

Theorem 2.6. If g(y) is a function defined on I such that y2β−1g(y) ∈ L(I), if
g(y) is of bounded variation in a neighborhood of a point y = x ∈ I and if G∗(y) is
given by (2.17), then∫ ∞

0

y−4αb∗α−β(xy)G
∗(y)dy =

1

2
[g(x + 0) + g(x− 0)],

for any real value of (α−β).

By an application of Fubini’s theorem, we can establish another Parseval equa-
tions involving a pair of these four transformations Hi,α,β(i = 1, 2, 3, 4).

Theorem 2.7. Let (α−β) ≥ 0.
(a) If x2αf(x) ∈ L(I) and y2β−1G3(y) ∈ L(I), then

(2.18)

∫ ∞
0

f(x)g(x)dx =

∫ ∞
0

F1(y)G1(y)dy,

where F1(y) = (H1,α,βf)(y) and G3(y) = (H3,α,βg)(y).
(b) The pair of transformations H2,α,β and H4,α,β verifies also the mixed Parseval
equation

(2.19)

∫ ∞
0

f(x)g(x)dx =

∫ ∞
0

F2(y)G4(y)dy

where now F2(y) = (H2,α,βf)(y) and G4(y) = (H4,α,βg)(y), provided that x2βf(x) ∈
L(I) and y−2βG4(y) ∈ L(I).



THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS 87

Remark 1. Note that ∆3,α,β and ∆4,α,β are formally the adjoints operators ∆1,α,β
and ∆2,α,β respectively.
By this reason, H3,α,β will be called the adjoint or complementary transform of
H1,α,β and we shall refer to H4,α,β as the adjoint or complementary transform of
H2,α,β.

Remark 2. The Parseval equality (2.18) involves the transformation H1,α,β and
its complementary H3,α,β where as the Parseval formula (2.19) relates both trans-
forms H2,α,β and H4,α,β. Observe that, in comparison with (2.2), (2.6), (2.11) and
(2.15), expressions (2.18) and (2.19) do not contain any weight function.

Remark 3. According to the notation we have just adopted, Theorem 2.7 admits
this statement more concisely.

Theorem 2.8. Let (α−β) be any fixed real number. If x2αf(x) and y2β−1G∗(y) ∈
L(I) and if we put F(y) = (Hα,βf)(y) and g(x) = (H∗−1α,β G

∗)(x) = (H∗α,βG
∗)(x),

then ∫ ∞
0

f(x)g(x)dx =

∫ ∞
0

F(y)G∗(y)dy,

that is,

(2.20)

∫ ∞
0

f(x)g(x)dx =

∫ ∞
0

(Hα,βf)(y)(H∗α,βg)(y)dy.

3. The generalized Schwartz’s Hankel Transformation of arbitrary
order:

Let (α − β) be any fixed real number and i an integer, 1 ≤ i ≤ 4. The space
Hi,α,β consists of all infinitely differentiable complex-valued functions φ(x) defined
on I for which

η
α,β,i
m,k (φ) = sup

x∈I

∣∣xm(x−1D)kwi(x)φ(x)∣∣
exist for each pair of non-negative integers m and k, where w1(x) = 1, w2(x) =
x−2(α−β), w3(x) = x

−4α and w4(x) = x
−1 with the topology generated by the

collections of seminorms
{
η
α,β,i
m,k

}
,Hi,α,β are Frechet spaces.

We denote H1,α,β = H1 and H4,α,β = H4, since the definition of these spaces is
independent of the particular choice of the parameter α−β.

Theorem 3.1. A function φ(x) defined on I is a member of Hi,α,β if and only if,
(a) φ(x) is infinitely differentiable on I,
(b) φ(x) has the form

φ(x) = w−1i (x)[a0 + a1x
2 + a2x

4 + · · · · · · + akx2k + o(x2k)]
in some vicinity of the origin, and
(c) Dkφ(x) is of rapid descent when x → ∞ for each k = 0, 1, 2, 3, . . . . . .. H′i,α,β
symbolizes the dual space of Hi,α,β and its members are generalized functions of
slow growth. We assign to H′i,α,β the weak topology generated by the multinorm

{ξi,φ} defined through
ξi,φ(f) =

∣∣〈f,φ〉∣∣ , f ∈ H′i,α,β, φ ∈ Hi,α,β.
Thus, H′i,α,β is also sequentially complete.



88 WAPHARE AND GUNJAL

Now we introduce the new spaces

(3.1) Hα,β =

{
H1, if (α−β) ≥ 0
H2,−α,−β if (α−β) ≤ 0

,

and

(3.2) H∗α,β =

{
H3,α,β, if (α−β) ≥ 0
H4, if (α−β) ≤ 0

.

H′α,β and H
∗′
α,β, denote the dual spaces of Hα,β and H

∗
α,β respectively some prop-

erties of these spaces are listed below:
(a) D(I) ⊂ Hi,α,β and the restriction of every f ∈ H′i,α,β to D(I) is a member of
D′(I). Moreover, Hi,α,β is a dense subspace of E(I).
Therefore, E′(I) is a subspace of H′i,α,β. Consequently, D(I) ⊂ Hα,β ⊂ E(I) and
D(I) ⊂ H∗α,β ⊂ E(I).
(b) The operations φ → ∆α,βφ and ψ → ∆∗α,βψ, where ∆α,β and ∆

∗
α,β represent

the operators given by (2.7) and (2.16), are continuous linear mappings from the
testing function spaces Hα,β and H

∗
α,β into themselves, respectively. Infact, the

inequality

η
α,β,i
m,k (∆i,α,βφ) ≤ |2k + 6α + 2β|η

α,β,i
m,k+1(φ) + η

α,β,i
m+2,k+2(φ),

i = 1, 2, 3, 4, is satisfied for every φ ∈ Hi,α,β by taking into account that

∆i,α,β = w
−1
i (x)[x

2(x−1D)2 + (6α + 2β)(x−1D)]wi(x).

Hence, the generalized differential operator ∆α,β defined on the distributional space
H∗α,β as the adjoint operator of ∆

∗
α,β, that is to say,〈

∆α,βf,ψ
〉

=
〈
f, ∆∗α,βψ

〉
, f ∈ H∗α,β, ψ ∈ H

∗
α,β,

is also a continuous linear mapping of H∗α,β into itself. Conversely, the generalized

differential operator ∆∗α,β will be defined on H
′
α,β by means of〈

∆∗α,βf,ψ
〉

=
〈
f, ∆α,βψ

〉
, f ∈ H′α,β, ψ ∈ Hα,β

and produces a continuous linear mapping of the space H′α,β into itself.

(c) Assume that (α−β) is any real number. Then, Hα,β may be identified with a
subspace of H∗

′

α,β, that is, Hα,β ⊂ H
∗′
α,β, the topology of Hα,β being stronger than

that induced on it by H∗
′

α,β. Indeed, any f ∈ H1 generates a regular distribution f
in the space H′3,α,β by

(3.3)
〈
f,φ
〉

=

∫
I

f(x)φ(x)dx, φ ∈ H3,α,β.

provided that (α −β) ≥ 0. The linearity of (3.3) being obvious, the continuity is
inferred from ∣∣〈f,φ〉∣∣ ≤ ηα,β,30,0 (φ) ∫

I

x4α|f(x)|dx.

Notice that last integral exists since f(x) = o(1) when x → 0, and f(x) is of rapid
descent at infinity, by virtue of Theorem 3.1. To see the second part of the assertion,



THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS 89

observe that

ξ3,φ(f) ≤ ρ
α,β,1
0,0 (f)

∫ ∞
0

|φ(x)|dx.

On the other hand, two members f and g of H1 giving rise to the same regular
distribution in H′3,α,β must be identical.

These considerations justify the inclusion H1 ⊂ H′3,α,β. Analogously, H2,−α,−β may
be identified one-to-one with a subspace of H′4. In other words, H2,−α,−β ⊂ H′4
whenever (α−β) ≤ 0. We can now conclude, in view of notation (3.1) and (3.2),
that Hα,β ⊂ H∗α,β. Finally, we can proceed in similar way to get the inclusions
H3,α,β ⊂ H′1 and H4 ⊂ H′2,−α,−β for (α − β) ≥ 0 and (α − β) ≤ 0, respectively,
from which one deduces that H∗α,β ⊂ H

′
α,β.

Theorem 3.2. (a) The Hankel type transformation Hα,β defined by (2.8) is an
automorphism on the space Hα,β, no matter what the real value of (α−β) be.
(b) The complementary Hankel type transformation H∗α,β, as given by (2.17) is as
well an automorphism on H∗α,β whatever be the real number α−β.

Proof. (a) In [1, Th.5] it was proven that H1,α,β is an automorphism on H1. To
study the transformation H2,α,β, firstly we note that the operation φ → x2(α−β)φ
is an isomorphism from H1 into H2,α,β and then we take into account the relation
(2.5) between transformations H1,α,β and H2,α,β.
(b) Since the operation ψ → x4αψ is an isomorphism from H1 into H3,α,β the
expression

(H3,α,βφ)(y) = y2α
[
H1,α,β(x−4αφ)

]
(y), φ ∈H3,α,β,

which connects the transforms (2.9) and (1.2), implies that H3,α,β is an automor-
phism on H3,α,β. Finally in view that the mapping φ → xφ defines an isomorphism
from H1 into H4 and (2.14), we deduce that H4,α,β is an automorphism on the
space H4 as well. According to convention (2.17), we can now conclude the validity
of statement (b).

�

Let (α − β) be any fixed real number. We define the generalized Schwartz’s
Hankel transformation Hα,β of arbitrary order in the distributional space H∗

′

α,β, as
the adjoint operator of H∗α,β acting on H

∗
α,β, namely,

(3.4)
〈
H

′

α,βf,φ
〉

=
〈
f,H∗α,βφ

〉
,

for every f ∈ H∗
′

α,β and Φ ∈ H
∗
α,β. If we put φ = H

∗
α,βΦ, the application of part (b)

in Theorem 3.2 allows us to rewrite (3.4) as follows

(3.5)
〈
Hα,βf,H∗α,βφ

〉
=
〈
f,φ
〉
, f ∈ H∗

′

α,β, φ ∈ H
∗
α,β.

This expression can be understood as an extension of the Parseval equation (2.20)
to distributions.

Theorem 3.3. The generalized Schwartz’s Hankel type transformation H′α,β given

by (3.4) or (3.5) is an automorphism on H∗
′

α,β, whatever be the real value of the
parameter α−β.

Proof. It is a consequence of part (b) in Theorem 3.2 and [17, Th.1.10-2]. �



90 WAPHARE AND GUNJAL

Remark 4. Note that the generalized transformation H
′

α,β is defined on the space

H∗
′

α,β as the adjoint operator of the complementary transform, and not over the

space H
′

α,β by means of the adjoint operator of itself, as usual in the available lit-

erature. This is suggested and justified, at once, by the inclusion Hα,β ⊂ H∗
′

α,β.

Indeed, because of this inclusion if Hα,β acts on Hα,β, the most natural would
be to define Hα,β on H∗

′

α,β instead of doing it on the space H
′
α,β.

On the other hand, when H
′

α,β is defined as usual, that is,

(3.6)
〈
H

′

α,βf, Φ
〉

=
〈
f,Hα,βΦ

〉
, f ∈ H

′

α,β, Φ ∈ Hα,β.

A Schuitman [10] found the relation

H
′

α,βf = y
4αHα,β(x−4αf), f ∈ Hα,β,

between classical and generalized transformations, which means that the conven-
tional transformation Hα,β is not a particular case of generalized one.
On the contrary, if definitions (3.4) or (3.5) are adopted, f ∈ Hα,β implies that
Hα,βf ∈ Hα,β in view of part(i) in Theorem 3.2.
Therefore, Hα,βf generates a regular member in H∗

′

α,β through (3.3),

(3.7)
〈
Hα,βf, Φ

〉
=

∫
I

(Hα,βf)(y)Φ(y)dy, Φ ∈ H∗α,β.

By virtue of Parseval relation (2.20), the right-hand side of (3.7) takes the form∫
I

f(x)
(
H∗α,β(Φ)

)
(x)dx =

〈
f,H∗α,βΦ

〉
=
〈
H

′

α,βf, Φ
〉
,

by definition (3.4). We have〈
Hα,βf, Φ

〉
=
〈
H

′

α,βf, Φ
〉
,

for all Φ ∈ H∗α,β. In other words, the classical transformation Hα,β coincides now
with the generalized one Hα,β, when this acts on the testing-function space H∗α,β.
This proves that the definition (3.4) is more appropriate than (3.6).

Remark 5. Note that the definitions of the generalized operators introduces in
(ii) are consistent with the rules of distributional calculus, in particular, with the
definitions of multiplication by an infinitely differentiable function, differentiation
of distribution which does not occur in [1], [5] and [10].

For every real α − β, it is easily seen that ∆α,β acts as a classical differential
operator on the space Hα,β and as a generalized one on H∗

′

α,β, while that the con-
ventional operator ∆∗α,β acting on H

∗
α,β is understood as a distributional operator

on H′α,β. These facts are justified in view of the inclusions Hα,β ⊂ H
∗′
α,β and

H∗α,β ⊂ H
′
α,β, since ∆

∗
α,β, is the adjoint operator of ∆α,β.



THE COMPLEMENTARY HANKEL TYPE TRANSFORMATIONS 91

Remark 6. Assume that (α − β) is any real number. The generalized transfor-
mation H

′

α,β can be similarly defined on the space H
′
α,β as the adjoint operator of

Hα,β on Hα,β, namely,

(3.8)
〈
H

′

α,βg, Ψ
〉

=
〈
g,Hα,βΨ

〉
, g ∈ H

′

α,β, Ψ ∈ Hα,β.

Finally, set ψ = Hα,βΨ. Then, because of Theorem 3.2(a), (3.8) becomes〈
H

′

α,βg,Hα,βΨ
〉

=
〈
g,ψ

〉
, g ∈ H

′

α,β, ψ ∈ Hα,β.

Again this expression can be considered as an exact transcription to generalized
functions of the mixed Parseval relation (2.20). It is fulfilled that H

′

α,βg = Hα,βg
whatever be g ∈ H∗α,β as well.

An analogous result to Theorem 3.3 can be inferred from Theorem 3.2(a):

Theorem 3.4. The generalized complementary Hankel transformation H
′

α,β, as

given by (3.8) is an automorphism on H
′

α,β independently of the value of (α−β) ∈
R.

Next, we collect some operation-transform formulas.

Theorem 3.5. Let (α−β) be an arbitrary real number. For all φ ∈ Hα,β,

Hα,β{∆α,βφ} = −y2Hα,βφ
∆α,βHα,βφ = Hα,β(−x2φ).

If ψ ∈ H∗α,β, then

Hα,β{∆∗α,βψ} = −y
2Hα,βψ

∆∗α,βHα,βφ = Hα,β(−x
2φ).

For every f ∈ H∗
′

α,β one has

Hα,β{∆α,βf} = −y2Hα,βf

whereas

H
′

α,β{∆
∗
α,βg} = −y

2H
′

α,βg

holds for any g ∈ H′α,β.

Remark 7. The Cauchy problem posed by W.Y. Lee [5]

∂u

∂t
= P(∆α,β,x)u(x,t)

u(x, 0) = f(x)

where f(x) is a known member of H∗
′

α,β, P denotes a polynomial with constant

coefficients and the unknown function u(x,t) ∈ H∗
′

α,β, can now be solved by applying

the transformation Hα,β not only for (α−β) ≥−12 , but whatever be the real value
of the parameter α−β. This and above results justify that the transformation Hα,β
be called the generalized Schwartz Hankel type transformation of arbitrary order.



92 WAPHARE AND GUNJAL

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