Abstract
The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. A proper subspace can be identified with the space of distributions. In this paper, we first construct a suitable Boehmian space on which the Sumudu transform can be defined and the function space S can be embedded. In addition to this, our definition extends the Sumudu transform to more general spaces and the definition remains consistent for S elements. We also discuss the operational properties of the Sumudu transform on Boehmians and finally end with certain theorems for continuity conditions of the extended Sumudu transform and its inverse with respect to δ and Δconvergence.
MSC: 54C40, 14E20, 46E25, 20C20.
Keywords:
Sumudu transforms; Boehmian spaces; the space ; the space ; distributions1 Introduction
The Sumudu transform of one variable function is introduced as a new integral transform by Watugala in [1] and is given by
over the set of functions
where is a function that can be expressed as a convergent series [2,3]. The Sumudu transform was applied to solve the ordinary differential equations in control engineering problems; see [3].
The Sumudu transform of the convolution product of f and u is given by
where and are the Sumudu transforms of f and u, respectively.
Some of the properties were established by Weerakoon in [4,5]. In [6], further fundamental properties of this transform were also established by Asiru. Similarly, this transform was applied to a onedimensional neutron transport equation in [7] by Kadem.
In [8], the Sumudu transform was extended to the distributions and some of their properties were also studied. Recently, this transform has been applied to solve the system of differential equations; see Kılıçman et al. in [9].
Note that a very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except the factor n; see Zhang [10]. Similarly, the Sumudu transform sends combinations into permutations , and hence it will be useful in the discrete systems.
The following are the general properties of the Sumudu transform which are auxiliary from the substitution method and the properties of integral operators.
(i) Ifandare nonnegative integers andandare the corresponding Sumudu transforms ofand, respectively, then
(iii) , whereis the Sumudu transform off.
More properties of the Sumudu transforms a long with a some of applications were given in [11] and [12].
2 Boehmian space
Boehmians were first constructed as a generalization of regular Mikusinski operators [13]. The minimal structure necessary for the construction of Boehmians consists of the following elements:
(ii) a commutative semigroup ;
(iii) an operation such that for each and , ;
Elements of Δ are called delta sequences. Consider
Now if , , , then we say . The relation ∼ is an equivalence relation in ℚ. The space of equivalence classes in ℚ is denoted by β. Elements of β are called Boehmians.
We note that between and β there is a canonical embedding expressed as . The operation ⊙ can also be extended to by . The relationship between the notion of convergence and the product ⊙ is given by:
(i) If as in and is any fixed element, then in A (as );
(ii) If as in and , then in (as ).
The operation ⊙ can be extended to as follows: Ifand, then. In β, there are two types of convergence as follows.
(1) A sequenceinβis said to beδconvergent tohinβ, denoted by, if there existssuch that, , andasinfor every.
(2) A sequenceinβis said to be Δconvergent tohinβ, denoted by, if there exists asuch that, , andasin.
3 The Boehmian space
Denote by and the space of all rapidly decreasing functions over () and the space of all test functions of compact support, respectively. In what follows, we obtain preliminary results required to construct the Boehmian space , where .
Lemma 3.1
Proofs are analogous to those of classical cases and details are omitted.
Definition 3.2 A sequence of functions from is said to be in if and only if
This means that shrinks to zero as . Each member of is called a delta sequence or an approximate identity or, sometimes, a summability kernel. Delta sequences, in general, appear in many branches of mathematics, but probably the most important applications are those in the theory of generalized functions. The basic use of delta sequences is the regularization of generalized functions, and further, they can be used to define the convolution product and the product of generalized functions.
Theorem 3.5Letandsuch that, , thenin.
Proof We show that in . Let K be a compact set containing the for every . Using , we write
The mapping , where , is uniformly continuous from . From the hypothesis that as (by ), we choose such that for large n and . This implies
Hence using and (3.2), (3.1) becomes
Thus in . Similarly, we show that . This completes the proof of the theorem. □
Proof
In view of the hypothesis of the theorem, we write
The last equation follows from the fact that [17]
The proof of the theorem is completed. □
Proof
In view of the analysis employed for Theorem 3.5, we get
Hence
This completes the proof. The Boehmian space is therefore constructed. □
The canonical embedding between and is expressed as . The extension of ⋆ to is given by . Convergence in is defined in a natural way:
δconvergence: A sequenceinis said to beδconvergent tohin, denoted by, if there exists a delta sequencesuch that, , andasinfor every.
convergence: A sequenceinis said to beconvergent tohin, denoted by, if there exists asuch that, , andasin.
Theorem 3.8The mappingis a continuous embedding ofinto.
Proof The mapping is onetoone. For detailed proof, let , then . Then since , . Using Theorem 3.5, we get . To show the mapping is continuous, let as in . Then we show that
From Theorem 3.5, as . This completes the proof of the theorem. □
4 The Boehmian space
We describe another Boehmian space as follows. Let be the space of rapidly decreasing functions [17]. Define
where denotes the Sumudu transform of u. We also define by
Proof If and , then using the topology of and Leibnitz’ theorem, we get
where and K is a compact subset containing the . Hence
for some positive constant M. This completes the proof of the lemma. □
Lemma 4.2The mapping
satisfies the following properties:
Proof The proof of the above lemma is straightforward. Detailed proof is as follows.
Proof of (1). Let , then . Hence by (4.1). Theorem 3.9 implies .
Proof of (2) is obvious.
Proof of (3). We have
that is,
This completes the proof of the theorem. □
Denote by the set of all Sumudu transforms of delta sequences from . That is,
Lemma 4.3Let, be such that, ∀n, thenin.
Proof Let and . Since , (4.2) implies . Hence for all x. The proof is completed. □
Proof Since , for all n. Hence, from Theorem 3.9, we get for every n. This completes the proof of the lemma. □
By aid of Lemma 4.3. and Lemma 4.4, can be regarded as a delta sequence.
Proof It is clear that is bounded in . Further,
Proof Let , then uniformly on compact subsets of . Hence
as . Thus as . This yields in the topology of . The proof is therefore completed. The space can be regarded as a Boehmian space, where . □
Lemma 4.7The mapping
is a continuous embedding ofinto.
Proof For , , is a quotient of sequences in the sense that . We show that the map (4.4) is onetoone. Let , then , . Using Lemma 4.2 and Lemma 4.3, we conclude . □
To establish the continuity of (4.4), let as in . Then as by Lemma 4.6, and hence
5 The Sumudu transform of Boehmians
Let , then we define the Sumudu transform of β by the relation
Proof Let , where , . Then the concept of quotients yields . Employing Theorem 3.9, we get , i.e., . Equivalently, . Thus . This completes the proof of the theorem. □
Theorem 5.2is continuous with respect toδconvergence.
Proof Let in , then by [14], and as in . Applying the Sumudu transform to both sides yields as . Hence
as in . This proves the theorem. □
Theorem 5.3is a onetoone mapping.
Since the Sumudu transform is onetoone, we get . Thus
Hence
This completes the proof of the theorem. □
Proof is immediate from the definitions.
Theorem 5.5is continuous with respect toconvergence.
Proof Let in as . Then there exist and such that and as . Employing Eq. (5.1), we get
Hence, we have as in . Therefore
Proof Let be arbitrary, then for every . Then . That is, is the corresponding quotient of sequences of . Thus is such that in . This completes the proof of the lemma.
Let , then we define the inverse Sumudu transform of by
Proof is immediate from the definitions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both of the authors contributed equally to the manuscript and read and approved the final draft.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.
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