International Scholarly Research Notices

International Scholarly Research Notices / 2014 / Article

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Volume 2014 |Article ID 463901 | https://doi.org/10.1155/2014/463901

S. K. Q. Al-Omari, "On the Sumudu Transform and Its Extension to a Class of Boehmians", International Scholarly Research Notices, vol. 2014, Article ID 463901, 6 pages, 2014. https://doi.org/10.1155/2014/463901

On the Sumudu Transform and Its Extension to a Class of Boehmians

Academic Editor: C. Zhu
Received19 Jan 2014
Accepted25 Feb 2014
Published24 Apr 2014

Abstract

Boehmians are used for all objects obtained by an algebraic construction similar to that of the field of quotients. In literature, several integral transforms have been extended to various Boehmian spaces but a few to the space of strong Boehmians. As shown in the work of Al-Omari (2013), this work describes certain spaces of Boehmians. The Sumudu transform is therefore established and it is one-one and continuous in the space of Boehmians. The inverse transform is given and some results are also discussed.

1. Introduction

Integral transforms are widely used in the literature, where some are often used for solving differential equations. The Sumudu transform was introduced by Watugala [1] and discussed by Weerakoon in [2]. Many properties are established in [35]. Having scale and unit-preserving properties, the Sumudu transform can be used to solve problems without resorting to new frequency domains. In [6] the classical Sumudu transform has been investigated over functions of two variables. In [7] the classical theorems in [6] are extended to distribution spaces and a space of Boehmians.

In this note we discuss the cited transform on certain space of strong Boehmians. Definitions and classical properties of Sumudu transforms are briefly given in Section 1. The general construction of usual and strong Boehmians is given in Sections 2 and 3, respectively. In Section 4, the Sumudu transform is extended to Boehmians and many properties are also obtained.

The Sumudu transform of is given by [1] over a set of functions given as where is of convergent series type and .

Denote by the usual convolution product of and .

Then the Sumudu transform of the convolution product is given as where and denote the Sumudu transform of and , respectively.

Following are general properties of Sumudu transforms.(i)If , and , are the Sumudu transforms of and , respectively, then (ii) (iii)

For details, see [8].

For strong Boehmians, see [9]. For usual Boehmians, see [1019].

2. Strong Boehmians

Denote by the set of positive real numbers. By denote the Schwartz space of test functions of compact support over and by the space of smooth functions over , where , . The dual space of , namely , consists of distributions of compact support [14, 20, 21].

Between and , the convolution product of and is given by where .

Since notations and denote a same product, when is either defined over or , the role of and may be interchanged in the text of this paper.

Let be the subset of of test functions such that The pair of functions, , , is said to be a quotient of functions, denoted as , if and only if where Two quotients and are said to be equivalent, , if and only if

Let the set be given as Then the equivalence class containing is said to be a strong Boehmian.

The space of all such Boehmians is denoted by and is the so-called the space of strong Boehmians.

Following conclusions are useful in the sequel [9, p.p. ].( )Let ; then .( )Let and ; then .( )Let and ; then we have that and ( )Let ; then .( )Let , and ; then we have where .

Addition and scalar multiplications in are defined in the usual way as Differentiation and the operation in the space in are given as

Convergence, in , is defined as follows. A sequence of strong Boehmians is said to be convergent to a Boehmian in if for some , , and such that , , we have as in .

3. General Boehmians

Boehmians were first constructed as a generalization of regular Mikusinski operators [16]. The minimal structure necessary for the construction of Boehmians consists of the following elements: (i) A set ; (ii) a commutative semigroup ,* ; (iii) an operation such that for each and , , a collection , ; a collection such that if , ,    , then ; if , then .   is the set of all delta sequences. Consider If , , , then we say . The relation is an equivalence relation in . The space of equivalence classes in is denoted by . Elements of are general Boehmians. Between and there is a canonical embedding expressed as . The operation can be extended to by . In , there are two types of convergence. ( ) A sequence in is said to be convergent to in , denoted by , if there exists a delta sequence such that ,    ,   , and as , in , for every . ( ) A sequence in is said to be convergent to in , denoted by , if there exists a such that , , and as in .

The following is equivalent for the statement of convergence:    in   if and only if there is and such that , and for each , as in .

For further discussion see [1019].

4. Sumudu Transform of Strong Boehmians

We start investigation by establishing the following theorem.

Theorem 1 (the convolution theorem). Let and . Then one has where , .

Proof. By using the definition of Sumudu transform, Fubini’s theorem then gives
The substitution implies and, hence
Hence
This completes the proof of the theorem.

Next, we are describing a usual space of Boehmians by images of Sumudu transforms of strong Boehmians.

Consider the following definition.

Definition 2. Let , or simply , be a set of delta sequences such that and ,   ,   as .
Let be the set of images of Sumudu transforms of all elements and, be the set of Sumudu transforms of all delta sequences from .
For and , we introduce the operation as

Lemma 3. Let and ; then , .
Proof is immediate since and .

Lemma 4. The following hold true. (i)If , then ,   .(ii)Let and be such that ; then

Proof. Proof of (i). For all we can find their related sequences . Since ,   , it follows , by (23).
This proves Part (i) of the lemma.
Proof of (ii). For each delta sequence we see that Hence
This implies
This completes the proof of the lemma.

Hence, from Lemma 4, can be regarded as delta sequence in .

Lemma 5. The mapping defined by
Satisfies the following properties. (i) , for every , .(ii)If and , then .(iii)If , , then .
Proof is straightforward from definitions.
For similar proofs see [15] and other cited papers.

Lemma 6. The following hold true. (i)If and then as .(ii)If and then as .

Proof. Proof of (i). Let , , . The hypothesis of the lemma implies
as in .
Proof of (ii). Let ; then the fact that as implies
as in .
This completes the proof of the lemma.

The general Boehmian space , or , is, therefore, constructed.

The sum and multiplication by a scalar of two Boehmians in are defined in a natural way as The operation and differentiation in are, respectively, defined by On the other hand, we are concerned with the strong space of Boehmians that can be described by the set and the subset , injected with the family of delta sequences. Such a space is denoted by , or, simply, by , for more convenience. With this injection, the space preserves the operations of addition, scalar multiplication, differentiation, and the convolution given to general Boehmians. Theorem 1 then suggests the following definition.

Definition 7. Let and ; then we define the Sumudu transform of the strong Boehmian in by where has its usual meaning.

Theorem 8. The Sumudu transform is well-defined.

Proof. Let be such that ; then using (12) we get Applying the convolution theorem on both sides of (35) yields Hence This completes the proof of the theorem.

Theorem 9. Let and ; then the mapping is one-one.

Proof. Assume in . Using (34) we get
By (23) we get
Using Theorem 1, (39) becomes
Since is one-one we get
The property of equivalence classes in , therefore, implies
Hence .
This completes the proof of the theorem

Theorem 10. is continuous with respect to convergence.

Proof. Let . By using convergence in and [9, Theorem 2.6] we can find common for all such that , and as . Hence as . Therefore
Hence, as in .
This completes the proof of the theorem.

Definition 11. Let in ; then we define the inverse of by the formula
in .

Theorem 12. The mapping is well-defined.

Proof. Let ; then . Using (23) we get
Theorem 1 implies . Hence
This completes the proof of the Theorem.

Theorem 13. The mapping is linear.

Proof. Let and ; then using (23) we write
Also, is obvious.
The proof is completed.

Theorem 14. The mapping is continuous with respect to convergence.

Proof. Let , , , and as . Applying the inverse Sumudu transform yields as . Thus as .
This completes the proof of the theorem.

If the family of delta sequences is not described in the strong Boehmian space we have considered in this note, it will be interesting to pay attention to the following remarks.

Useful Remark. (i) The injection defined by

is continuous and independent of the choice of .

(ii) The injection defined by

is continuous.

Proof. Proof of (i) is same as that of [9, Theorem 3.1].
Detailed proof of (ii) is as follows.
Suppose , , and as uniformly on compact subsets of . Then
as .
Employing (i) we get
That is
as . Therefore
as .
This completes the proof of the remark.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2014 S. K. Q. Al-Omari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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