Abstract
This paper investigates the transform on a certain space of generalized functions. Two spaces of Boehmians have been constructed. The transform
is extended and some of its properties are also obtained.
1 Introduction
Integral transforms are widely used to solve various problems in calculus, mechanics, mathematical physics, and some problems appear in computational mathematics as well.
In the sequence of these integral transforms, the Laplace - type integral transform, so-called transform, is defined for a squared and an exponential function
by David et al. (Citation2007), as
(1) The
transform is related to the classical Laplace transform by means of the following relationships :
(2) and
(3) Let f and g be Lebesgue integrable functions; then the operation
between f and g is defined by
The operation
is commutative, associative and satisfies the equation
.
Some facts about the transform are given as follows:
| (1) |
| ||||
| (2) |
| ||||
| (3) |
| ||||
More properties, applications and the inversion formula of transform are given by Yürekli (Citation1999a,Citationb)".
2 Abstract construction of Boehmians
The minimal structure necessary for the construction of Boehmians consists of the following elements :
| (i) | A set | ||||||||||||||||
| (ii) | A commutative semigroup | ||||||||||||||||
| (iii) | An operation | ||||||||||||||||
| (vi) | A collection
| ||||||||||||||||
ConsiderIf
, then we say
. The relation
is an equivalence relation in
. The space of equivalence classes in
is denoted by
. Elements of
are called Boehmians.
Between and
there is a canonical embedding expressed as
The operation
can be extended to
by
The sum of two Boehmians and multiplication by a scalar can be defined in a natural way
and
The operation
and the differentiation are defined by
and
In particular, if
and
is any fixed element, then the product
, defined by
is in
.
Many a time is also equipped with a notion of convergence. The intrinsic relationship between the notion of convergence and the product
is given by:
If
In , two types of convergence are:
| (1) | A sequence | ||||
| (2) | A sequence | ||||
The following theorem is equivalent to the statement of - convergence :
Preposition 1
in
if and only if there is
and
such that
and for each
as
in
.
For further discussion of Boehmian spaces and their construction; see Ganesan (Citation2010), Karunakaran and Ganesan (Citation2009), Al-Omari (Citation2012, Citation2013a,Citationb,Citationc,Citationd), Al-Omari and Kilicman (Citation2011, Citation2012a,Citationb, Citation2013a,Citationb), Boehme (Citation1973), Bhuvaneswari and Karunakaran (Citation2010), Ganesan (Citation2010), Karunakaran and Angeline (Citation2011), Karunakaran and Devi (Citation2010), Mikusinski (Citation1983, Citation1987, Citation1995)", Nemzer (Citation2006, Citation2007, Citation2008, Citation2009, Citation2010)" and Roopkumar (Citation2009).
3 The Boehmian space
denotes the space of rapidly decreasing functions defined on
. That is,
if
is a complex-valued and infinitely smooth function defined on
and is such that, as
and its partial derivatives decrease to zero faster than every power of
.
In more details, iff it is infinitely smooth and is such that
(4) m and k run through all non-negative integers; see Pathak (Citation1997).
denotes the Schwartz space of test functions of bounded support defined on
.
denotes the Mellin-type convolution product offirst kind defined by Zemanian (Citation1987), as
(5) To construct the first Boehmian space
, we need to establish the following necessary theorems.
Theorem 2
Let and
; then we have
.
Proof
Let be a compact subset in
containing the support of
. Then, for all
and
, we, by Equation(4)
(4) and Equation(5)
(5) , get that
Hence, by considering the supremum over all
, we obtain that
.
This completes the proof of the theorem. □
Theorem 3
Let ; then we have
| (1) |
| ||||
| (2) |
| ||||
| (3) |
| ||||
Proof of (1) and (3) follows from the Reference (Pathak, Citation1997). Proof of (2) is straightforward from the properties of integration. Therefore, we prefer to omit the details.
Hence the theorem is completely proved.
Theorem 4
Let in
as
and
; then
as
.
Proof of this theorem is straightforward from simple integration.
Definition 5
Denoted by the set of all sequences
that satisfying:
| |||||
| |||||
| |||||
Each is called a delta sequence or an approximating identity which corresponds to the Dirac delta distribution.
Theorem 6
Let and
; then
as
.
Proof
By Axiom of Definition 5, we write
(6) The mapping
is uniformly continuous for each
. Therefore, it follows that
as
.
Hence, we have established that as
.
Hence the theorem has been proved. □
Now, we assert that the product of delta sequences is a delta sequence. Detailed proof is as follows.
Let . Then, we have
Next, let
be constants such that
and
; then
Once again, change of variables, implies
Finally, the inequality
establishes our assertion.
The space is therefore considered as a space of Boehmians.
Definition 7
Let and
. Between
and
define a product
as in the integral equation
(7) We establish the second Boehmian space
.
Theorem 8
Let and
then we have
.
Proof
By Equation(7)(7) , we write
(8) Since
, we, from Equation(8)
(8) , find that
(9) where
is a compact set in
containing the support of
.
Hence, Equation(9)(9) leads to the conclusion that
.
This completes the proof of the theorem. □
Theorem 9
Let and
then
.
Proof
Let the hypothesis of the theorem satisfies for and
Then, using Equation(5)
(5) and Equation(7)
(7) yield
(10) The change of variables
implies
.
Hence, from Equation(10)(10) , we obtain
This completes the proof of the theorem. □
Theorem 10
Let and
then we have
| |||||
| |||||
Theorem 11
Let in
as
; then
as
, for each
.
Proof of this theorem follows from Theorem 9.
Theorem 12
Let and
; then
as
.
Proof
Utilizing Definition 5 and Axiom yield
(11) Since
it follows
Hence, fromEquation(11)
(11) and Definition
, we get
Hence the theorem is therefore completely proved. □
The Boehmian space has been constructed.
Theorem 13
Let and
then we have
Proof
Let and
; then we have
The substitution
implies
Hence the theorem is proved. □
In view of above, we define the extended transform of a Boehmian
in
as
(12)
Theorem 14
is well-defined.
Proof
Let in
. Then, by the concept of quotient of sequences in
, we write
Employing
on both sides of the above equation implies
That is,
Therefore, we get
This completes the proof of the theorem. □
It is also interesting to know that the transform is a linear mapping from
into
.
Detailed proof can be given as follows : If then
Hence,
Also, if
, the field of complex numbers, then we see that
Hence,
This completes the proof of the theorem.
Definition 15
Let ; then we define the inverse
transform of
as
(13) for each
.
Theorem 16
is an isomorphism.
Proof
Assume . Using Equation(12)
(12) and the concept of quotients we get
. Therefore, Theorem 13 implies
Properties of
implies
. Therefore, from the concept of qoutients of equivalent classes of
, we get
To establish that
is surjective, let
. Then
for every
. Once again, Theorem 13 implies
. Hence
is a Boehmian satisfying the equation
This completes the proof of the theorem. □
Theorem 17
Let and
then we have
Detailed proof of the first part is as follows : Applying Equation(13)(13) yields
By using Equation(12)
(12) , we obtain
The proof of the second part
is similar.
This completes the proof of the theorem.
Theorem 18
and
are continuous with respect to
and
-convergence.
Proof
First of all, we show that and
are continuous with respect to
-convergence.
Let in
as
; then we show that
as
. By virtue of Preposition 1 we can find
and
in
such that
and
as
for every
. Hence,
as
in the space
. Thus,
as
in
.
To prove the second part, let in
as
. Then, once again, by Preposition 1,
and
and
as
. Hence
in
as
. Or,
as
. Using Equation(13)
(13) we get
Now, we establish continuity of
and
with respect to
-convergence.
Let in
as
. Then, there exist
and
such that
and
as
.
Employing Equation(12)(12) gives
Hence, we have
as
in
.
ThereforeHence,
as
.
Finally, let in
as
; then we find
such that
and
as
for some
.
Now, using Equation(13)(13) , we obtain
Theorem 13 implies
Thus
From this we find that
as
in
.
This completes the proof of the theorem. □
Theorem 19
The extended transform is consistent with
.
Proof
For every , let
be its representative in
; then
, where
. Its clear that
is independent from the representative,
. Therefore
which is the representative of
in
.
Hence the proof is completed. □
Theorem 20
Let ; then the necessary and sufficient condition that
to be in the range of
is that
belongs to range of
for every
.
Proof
Let be in the range of
; then of course
belongs to the range of
.
To establish the converse, let be in the range of
. Then there is
such that
.
Since, ,
. Therefore,
where
and
.
The fact that is injective, implies that
.
Thus, is qoutient of sequences in
. Hence,
Hence the theorem is proved. □
Theorem 21
Let and
; then
Proof
Assume the requirements of the theorem are satisfied for some and
. Then
Therefore,
This completes the proof of the theorem. □
Notes
Peer review under responsibility of University of Bahrain.
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