Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
Hankel type transform on a Gevrey type space
Authors & Affiliation:
Abstract:
B.B.Waphare
MAEER’s MIT Arts, Commerce
& Science College, Alandi,
Pune, India.
In this paper we give a completion for the distributional theory of Hankel type
transformation developed in [6] and [8]. The space
generalizing the
Altenburg space is defined. Some properties of this space are studied. It is
shown that the Hankel type transformation ℎ , is an automorphism of
.
[email protected]
The generalized Hankel type transform of Gevrey spaces is defined and it is
found that Hankel transform is an automorphism of
. Product on
, Hankel
type translation and Hankel type convolution on
are investigated.
KEY WORDS:
Hankel type transform,
Hankel type translation,
Hankel type convolution,
Gevrey spaces..
Corresponding Author:
B.B.Waphare
© 2015.The Authors. Published
under
Caribbean Journal of
Science and Technology
ISSN 0799-3757
http://caribjscitech.com/
864
Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
2000 mathematics subject classification : 46 F 12, 46 F05.
1. Introduction : The space
, 1≤
,
< ∞, consists of all those measurable functions
on = (0, ∞)
such that
∞
‖ ‖
where
∞
,
,
= ∫ | ( )|
denotes the space of those functions
‖ ‖
,
= ess ℓ. u. b
∈
( )
⁄
< ∞,
(1.1)
for which
| ( )| < ∞.
(1.2)
.
(1.3)
Note that
( )=
)
Γ(
∈
The Hankel type transformation of
ℎ
,
( )= ∫
)
( ) and
∞
,
(
( ) is defined by
) ( )
( ),
∈ ,
(1.4)
where
( )=
( − ).
(
( ) represent the Bessel type function of the first kind and order
Throughout this paper we shall assume that ( − ) ≥ − 1 2 .
( ) is bounded on I, the Hankel type transform ℎ
Since ( )
(1.4) is given by
( )= ∫
∞
(
) ℎ
,
( )
( ),
,
( ) is bounded on I. The inversion for
∈ .
(1.5)
G. Altenburg [1] introduced the space that consist of all those complex valued and smooth function
defined on , such that for every , ∈ ℕ .
,
when
( )=
∈
(1 +
) |(
)
( )| < ∞,
is endowed with the topology associated with the family
(1.6)
,
, ∈ℕ
of seminorms,
is a Frechet
space and the Hankel type transformation ℎ , is an automorphism of , [1]. The Hankel type transform is
defined on , the dual space of , as the transpose of ℎ , on and is defined by ℎ , . That is if ∈
,
then the Hankel type transform ℎ ,
of is the element of
defined by
〈ℎ
If
∈
,
then
,
defines an element of
〈 , 〉= ∫
∞
( )
( )
, 〉 = 〈 ,ℎ
,
〉,
∈
.
through
( ),
∈
.
(1.7)
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Research Article
Thus,
,
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
. The convolution associated to the ℎ
can be seen as a subspace of
,
− transformation is
defined as follows:
The Hankel type convolution #
of order ( − ) of the measurable functions ,
,
∈
, ( )
is given
through
#
( )= ∫
,
∞
( )
where the Hankel type translation operator
( )= ∫
,
∞
of
,
( )
( , , )= ∫
∞
( ),
(1.8)
is defined by
( , , )
,
provided that the above integrals exist. Here
,
( )
,
( )
(1.9)
is the following function
,
( )
( )
( )
( )
(1.10)
( ).
(1.11)
and we have the following basic formula
∫
Lemma 1.1: Let
(i) ℎ
,
(ii) ℎ
,
∞
and
( )
be functions on
( )=
,
#
Proof: (i) As
(
( )= ℎ
,
( )=∫
,
( , , )
,
∞
) ℎ
( )
( ),
,
,
( )
( ), then
,
( ) ℎ
,
( )=
( ).
,
( , , )
( ).
We therefore have
∞
ℎ
,
,
( )=
( )
( )
,
( )
∞
∞
( )
=
= ∫
∞
( )
( )∫
∞
( )
( )
( )
,
( , , )
,
( , , )
( )
( ).
Using (1.11), we have
∞
ℎ
,
,
( )=
( )
(
)
(
)
( )
) ( )
( )
∞
=
(
)
(
866
Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
(
=
) ℎ
( ).
,
(1.12)
(ii) Proof follows from [5, p.339].
Thus proof is completed.
2. The Space
: Following G. Bjorck [4], we consider continuous, increasing and non-negative functions
defined on I, such that (0) = 0, ( ) > 0 and it satisfies the following three properties :
(i)
(ii)∫
( + )≤
∞
( )
( ) + ( ),
,
∈
(2.1)
< ∞
(2.2)
and
(iii)
( )≥
+ log(1 + ), for some
∈
> 0.
and
(2.3)
The class of all such functions is denoted by M. Note that if
satisfies the subadditivity property
(i) for every ,
is extended to ℝ as an even function then
∈ ℝ..
A. Beurling [3] developed the foundations of a general theory of distributions that extends the
Schwartz theory. Some aspects of that theory were presented and completed by Bjorck [4]. Now we collect
some definitions and properties for the purpose of the present paper.
A function
,
On
∈
( ) = Sup
,
∈
( ) is in
when
( )
)
|(
is smooth function and every ,
∈ ℕ ,
( )| < ∞ .
(2.4)
we consider the topology generated by the family
space. Following [2], we conclude that
,
of seminorms.
, ∈ℕ
is clearly a linear
is a Frechet space. For,
( ) = log(1 + ) it reduces to and for ( ) =
(0 < < 1),
is a Gevrey space. From definitions
( )
() ⊂
(1.6), (2.4) and the inequality (1 + ) ≤
, it follows that
⊆ H. It is clear that ( ) ⊂
( ). Since ( ) is a dense subspace of ( ), then
( ) is dense in ( ). Hence ( ) ⊂
( ) the dual of
( ). Since
⊂ the following properties given in [2,5] hold in the present case also when ∈ . The
Bessel type operator defined by
∆
,
=
=
+
4
.
By an application of Bessels equation for fixed, we have
∆
,
( )= −
Theorem 2.1: (i) The Hankel type transformation ℎ
(ii) The generalized Hankel type transformation ℎ
,
,
( ).
is an automorphism of
is an automorphism of
.
.
Proof: Here we prove (i) first and (ii) can be proved in a similar way.
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Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
As Hankel type transformation
∞
( )= ∫ (
,
is defined by
,
)
(
) ( )
,
∈ ,
(2.5)
we have
∞
ℎ
( )=
,
(
( )
∞
=
)
Γ(
∫ (
2
1
Γ(3 + )
2
1
Γ(3 + )
=
=
)
=
( )
)
(
)
)
( )
∞
(
)(
)
(
)(
)
( )
(
)(
)( )
∞
(
,
)
Γ(
(
)
(
)( )
(2.6)
Using relation (2.6), we have
(1 +
=
) (
Γ(
)
)
ℎ
(1 +
)
( )
,
(
)
(
,
).
Following technique of Zemanian [9, p.141], we can write
(1 +
=
Γ(
)
∫
)(
( ) (
(1 +
∞
×
)
(
)
(
)
(
)
ℎ
( )
,
)
)
(
.
)
From (ii) it follows that to every > 0, there exist a constant ( ) such that
( ) ≤
+ ( ) , so that
∞
( )
( )
<
( )
!
∞
( )
≤
( )
!
(1 +
ℎ
( )
) .
, we have
Now, for any choice and
,
ℎ
=
,
∞
( )
≤
∈
( )
!
( )
(1 +
(
)
)
(
,
) ℎ
,
( )
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Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
∞
( )
≤
∈
( )
!
1
Γ(3 + )
2
.
∞
(1 +
×
(
)
)
(
)
( ) (
)
(
)
(
)
Since
Γ(
)
(
)
(
)
(
) is bounded by
∞
,
ℎ
≤
,
( )
,
, for ( − ) ≥ − , so that
,
( )
!
(
(1 +
)
+ 1),
( )
)
∈
∞
× |(
)
∞
≤
( )
!
( )
,
( )|
−
( −
+
⁄2 +
+ 1),
⁄
( )
+
2+
< ∞,
≥ 1, by choosing
as infinite series can be made convergent for
<
1+
.
This proves that ℎ ,
is also in
and that ℎ , is continuous linear mapping from
into itself. Since
⊂ , ( ) for ( − ) ≥ − 1⁄2, we can apply inversion theorem and also the fact that ℎ , = ℎ , to
this case and conclude that ℎ , is an automorphism on
. Thus proof is completed.
Now the generalized Hankel type transformation ℎ
∈
More specifically, for any
〈ℎ
,
, 〉 = 〈 ,ℎ
,
∈
and
,
on
is defined as adjoint of ℎ
,
on
.
we have
〉.
3. Product, Hankel type translation and Hankel type convolution on I:
We shall denote by Λ the space of all
= ( ) ∈ ℕ for which
( ) |(
)
Proof: For ,
∈
function ( ) ,
∈ , and that
( )| < ∞
Here Λ is the space of multipliers for
Theorem 3.1: If ,
∞
( ), then
∈ ℕ and there exists a
(3.1)
.
∈
( ).
∈ ℕ , we have by definition (2.4)
,
(
) ( ) = Sup
( )
|(
)
( ) ( )|.
∈
Using Leibnitz theorem, we have
869
Research Article
,
(
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
)( )
( )
=
(
)
( )(
)
( )
(
)
∈
∞
( ) |(
=
)
( )|
∈
=
( )
,
( ) < ∞.
,
( ). This completes the proof of theorem.
∈
Hence
⟼
Theorem 3.2: The mapping
Proof: Let
( )
∈
( ). Then ℎ
∈
is continuous from
,
( ). By Lemma 1.1(i), we have
∈
,
ℎ
,
into itself.
( )=
,
(
) ℎ
( ).
,
Now we show that
(
) ∈ Λ .
We have
(
)
(
) = (
= (−1)
(
)
( )
(
(
Here
(
) ≤
)
(
(
(
)
, so that there exists
,
) < ∞ , for every
) ∈ Λ for fixed
∈ . But ℎ
, we have
an automorphism of
)
(
)
)
)
(
(
)
).
> 0 such that
∈ .
( ) ∈
,
∈
,
(
(
, then
⟼
and the mapping
)ℎ
( ) ∈
,
. Since ℎ
is continuous from
,
,
is
into
itself.
Theorem 3.3: If ,
( ) , then
∈
#
( ).
∈
,
Proof: By definition (2.4), we have
,
ℎ
#
,
( )
( )=
,
(
) ℎ
∈
,
#
,
.
Using Lemma 1.1 (ii), we have
,
ℎ
,
#
,
( )
( )=
∈
(
)
ℎ
,
( ) ℎ
,
( )
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Research Article
B.B.Waphare, Carib.j.SciTech, 2015, Vol.3, 864-871
( )
=
(
)
ℎ
( )
=
(
)
ℎ
∈
=
ℎ
,
Hence ℎ
,
#
,
( )
,
∈
,
ℎ
( )(
,
∈
( )
,
,
)
(
∈
ℎ
( )
,
) ℎ
,
( )
( ) < ∞.
( ). Since ℎ
,
is an automorphism of
( ), we have
#
,
∈
( ).
Thus proof is completed.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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Belhadj M and Betancor J.J., Hankel transformation and Hakel convolution of tempered Beurling
distributions, Rocky Mountain J. Math. 31(4) (2001), 1171-1203.
Beurling A. Quasi-analyticity and generalized distributions, Lecture 4 and 5,A.M.S. summer
Institute, Stanford, 1961.
Bjorck G., Linear partial differential operators and generalized distributions, Ark.Math. 6(21) (1966),
351-407.
Haimo D.T., Integral equations associated with Hankel convolution, Trans. Amer. Math. Soc. 116
(1965) ; 330-375.
B.B. Waphare, Pseudo-differential type operator on certain Gevrey type spaces,Global Journal of
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