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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 750524, 7 pages
http://dx.doi.org/10.1155/2013/750524
Research Article

Unified Treatment of the Krätzel Transformation for Generalized Functions

1Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan
2Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 26 March 2013; Accepted 18 April 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 S. K. Q. Al-Omari and A. Kılıçman. 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.

Abstract

We discuss a generalization of the Krätzel transforms on certain spaces of ultradistributions. We have proved that the Krätzel transform of an ultradifferentiable function is an ultradifferentiable function and satisfies its Parseval's inequality. We also provide a complete reading of the transform constructing two desired spaces of Boehmians. Some other properties of convergence and continuity conditions and its inverse are also discussed in some detail.

1. Introduction

Krätzel, in [1, 2], introduced a generalization of the Meijer transform by the integral: where , . Then this generalization is known as Krätzel transform.

Let be in . Denote by , or , the space of all complex-valued smooth functions on such that , where runs through compact subsets of ; see [3]. The strong dual of consists of distributions of compact supports.

Later, the authors in [4] have studied the transformation in a space of distributions of compact support inspired by known kernel method. They, also, have obtained its properties of analyticity and boundedness and have established its inversion theorem. In the sense of classical theory, the Meijer transformation and the Laplace transformation in [5] are presented as special forms of the cited transform for and , , respectively.

It is worth mentioning in this note that a suitable motivation of the cited transform has thoroughly been discussed in [6] by the aid of a Fréchet space of constituted functions of infinitely differentiable functions over .

This paper is a continuation of the work obtained in [4]. We are concerned with a general study of the transform in the space of ultradistributions and further discuss its extension to Boehmian spaces in some detail. We are employing the adjoint method and method of kernels for our purpose to extend the classical integral transform to generalized functions and hence ultradistributions.

2. Ultradistributions

The theory of ultradistributions is one of generalizations of the theory of Schwartz distributions; see [3, 7]. Since then, in the recent past and even earlier, it was extensively studied by many authors such as Roumieu [8, 9], Komatsu [10], Beurling [11], Carmichael et al. [12], Pathak [13, 14], and Al-Omari [15, 16].

By an ultradifferentiable function we mean an infinitely smooth function whose derivatives satisfy certain growth conditions as the order of the derivatives increases. Unlike sequences presented in [15, 16], , wherever it appears, denotes a sequence of positive real numbers. Such omission of constraints may ease the analysis.

Let be a real number but fixed and be the space of Lebesgue integrable functions on . Denote by (resp., ), , the subsets of of all complex valued infinitely smooth functions on such that, for some constant , for all (for some ), where is a compact set traverses .

The elements of the dual spaces,   , are the Beurling-type (Roumieu-type) ultradistributions. It may be noted that . Thus, every distribution of compact support is an ultradistribution of Roumieu type and further, and an ultradistribution of Roumieu type is of Beurling-type. Natural topologies on (resp., ) can be generated by the collection of seminorms: A sequence (resp., ) if and there is a constant independent of such that for all .      is dense in , convergence in    implies convergence in , and consequently a restriction of any to is in .

3. The Krätzel Transform of Tempered Ultradistributions

In this section of this paper we define the Krätzel transform of tempered ultradistributions by using both of kernel and adjoint methods. We restrict our investigation to the case of Beurling type since the other investigation for the Roumieu-type tempered ultradistributions is almost similar.

Lemma 1. Let and then .

Proof. Let and be fixed, and then certainly exists. By differentiation with respect to we get Hence the principle of mathematical induction on the th derivative gives From [4], we deduce that for some constant . The assumption that implies that the products under the integral sign, and , are also in . Moreover, ensures that the integral: belongs to . Hence for some constant . Therefore, from the above inequality we get for certain positive constant . This proves the lemma.

From Lemma 1 we deduce that the Krätzel transform is bounded and closed from into itself. Next, we establish the Parseval’s relation for the Krätzel transform.

Theorem 2. Let and be absolutely integrable functions over and then where and are the Krätzel transforms of and , respectively.

Proof. It is clear that and are continuous and bounded on . Moreover, the Fubini’s theorem allows us to interchange the order of integration: Equation (15) follows since the Krätzel kernel applies for the functions and , when the order of integration is interchanged. This completes the proof of the theorem.

Now, in consideration of Theorem 2, the adjoint method of extending the Krätzel transform can be read as where and .

Theorem 3. Given that then .

Proof. Consider a zero convergent sequence in then certainly is a zero-convergent sequence in the same space. It follows from (16) that
Linearity is obvious. This completes the proof.

From the above theorem we deduce that the Krätzel transform of a tempered ultradistribution is a tempered ultradistribution. Moreover, the boundedness property of follows from the following theorem.

Theorem 4. Let and then is bounded.

Proof. See [4, Proposition  2.3].

It is interesting to know that the Krätzel transform can be defined in an alternative way, namely, by the kernel method. Let , and then In fact, (18) is a straightforward consequence of Lemma 1.

Theorem 5. Let then is infinitely differentiable and for every and .

Proof. See [4, Proposition  2.2].

4. Boehmian Spaces

Boehmians were first constructed as a generalization of regular Mikusinski operators [17]. The minimal structure necessary for the construction of Boehmians consists of the following elements:(i)a nonempty set ,(ii)a commutative semigroup ,(iii)an aperation such that for each and ,(iv)a collection such that(a)if for all , then ,(b)if , then .

Elements of are called delta sequences. Consider If , , for all , 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 In , there are two types of convergence:( convergence)  a sequence in is said to be convergent to in , denoted by , if there exists a delta sequence such that , for all , and as , in , for every ,( convergence) a sequence in is said to be convergent to in , denoted by , if there exists a such that , for all , and as in . For further discussion see [1721].

5. The Ultra-Boehmian Space

Denote by , or , the Schwartz space of functions of bounded support. Let be the family of sequences such that the following holds:, for all ,, for all , as .It is easy to see that each in forms a delta sequence.

Let and be related by the expression: where for every .

Lemma 6. Let and and then .

Proof. Using the weak topology of , we write where . Hence, to complete the proof, we are merely required to show that .
First, if and , then choosing a compact set containing the support of yields which is dominated by . The dominated convergence theorem and the principle of mathematical induction implies Finally Therefore Thus . This completes the proof of the lemma.

Lemma 7. Let and and then(i), ,(ii).

Proof of the above Lemma is obvious.

Lemma 8. Let and and then

Proof. Let and then . Therefore Hence as .

Lemma 9. Let and then

Proof. Let and , and then It is sufficient to establish that as . By using (27) and imply that Hence, the mean value theorem implies . Let , where is certain compact set. The calculations show that where is certain constant. Hence Lemma 9. The Boehmian space is therefore constructed.

6. and the Krätzel Transform of Ultra-Boehmians

Denote by , or , the space of functions which are Krätzel transforms of ultradistributions in , and then convergence on can be defined in such away that in if as , where and . Let and and then it is proper to define for each .

Lemma 10. Let and and then .

Proof. Let be a compact set containing the support of , and then from (18) it follows that Hence the lemma follows.

Lemma 11. Let and and then .

Proof. implies , for some . Hence .

The following are lemmas which can be easily proved by the aid of the corresponding lemmas from the previous section. Detailed proof is avoided. First, if is the inverse Krätzel transform of , and then

Lemma 12. Let and and then

Lemma 13. Let and then for all we have(1), (2).

Lemma 14. Let and and then .

Lemma 15. Let and and then .

With the previous analysis, the Boehmian space is constructed. The sum of two Boehmians and multiplication by a scalar in is defined in a natural way and .

The operation and the differentiation are defined by

With the aid of Lemma 10 we define the extended Krätzel transform of a Boehmian to be a Boehmian in expressed by the relation:

Lemma 16. is well defined and linear mapping.

Proof is a straightforward conclusion of definitions.

Definition 17. Let and then the inverse of is defined as follows: for each .

Lemma 18. is an isomorphism from into .

Proof. Assume , then it follows from (41) and the concept of quotients of two sequences . Therefore, Lemma 10 implies . Employing properties of implies . Thus, . Next we establish that is onto. Let be arbitrary and then for every . Hence are such that . Hence the Boehmian satisfies the equation .
This completes the proof of the lemma.

Lemma 19. Let , , and and then

Proof. It follows from (42) that Applying Lemma 12 leads to Proof of the second part is similar. This completes the proof of the lemma.

Theorem 20. and are continuous with respect to and convergences.

Proof. First of all, we show that and are continuous with respect to convergence. Let in as and then we establish that as . In view of [10], there are and in such that such that as for every . The continuity condition of the Krätzel transform implies as , , and in the space . Thus, as in .
To prove the second part of the lemma, let as . From [10], we have and for some where as . Hence in as . That is, as . Using (42) we get as .
Now, we establish continuity of and with respect to convergence. Let in as . Then, we find , and such that and as . Employing (41) we get Hence, from (41) and Lemma 19 we have as in . Therefore Hence, as . Finally, let in as and then we find such that and as for some and .
Next, using (42), we obtain . Lemma 19 implies that Thus as . Hence, we have as in .
This completes the proof of the theorem.

Acknowledgment

The authors express their sincere thanks to the referees for the careful and detailed reading of the earlier version of paper and the very helpful suggestions that improved the paper substantially. The second author also acknowledges that this project was partially supported by University Putra Malaysia under the ERGS Grant Scheme with Project no. 5527068.

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