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Abstract and Applied Analysis

Volume 2014, Article ID 612391, 6 pages

http://dx.doi.org/10.1155/2014/612391
Research Article

On the Exponential Radon Transform and Its Extension to Certain Functions Spaces

1Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan

2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 25 October 2013; Accepted 18 April 2014; Published 5 May 2014

Academic Editor: Abdullah Alotaibi

Copyright © 2014 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 investigate the exponential Radon transform on a certain function space of generalized functions. We establish certain space of generalized functions for the cited transform. The transform that is obtained is well defined. More properties of consistency, convolution, analyticity, continuity, and sufficient theorems have been established.

1. Introduction

The Radon transform of a sufficiently nice function defined on is given by where is the unit sphere in , and is the Euclidean measure on the subspace orthogonal to .

Applications of the Radon transform occur in a number of areas, such as seismic signal processing, remote sensing, and system identification from output data [1, 2]. The Radon transform is extended to various spaces of distributions, rapidly decreasing and integrable Boehmians [3, 4]. More about the Radon transform is given in [59].

The discrete Radon transform is defined by [10, 11]. The attenuated Radon transform is defined in Mikusiński et al. [12, 13]. For a uniform attenuation coefficient , the exponential Radon transform of a compactly supported real valued function , defined on , is given by Kurusa and Hertle [7, 8]: where is a unit vector on , .

The exponential Radon transform constitutes a mathematical model for imaging modalities such as X-ray tomography for , single photon emission tomography for , and optical polarization tomography of trass tensor field [14]. However, if in addition is unknown, then one first must find and then find . This is the identification problem.

The exponential Radon transform, as a generalization of the Radon transform, is defined as a mapping of function spaces and is also represented in terms of Fourier transforms of its domain and range, and this leads to a characterization of the range of the transform. For more information about the exponential Radon transform, we refer to [15, 16].

2. General Construction of Boehmians

The minimal structure necessary for the construction of Boehmians consists of the following elements:(i)a set and a commutative semigroup ;(ii) an operation such that for each and , (iii) a collection satisfying the following:(1)if , , for all , then ;(2)if , then being 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 called Boehmians.

Between and there is a canonical embedding expressed as

The operation can be extended to by In , two types of convergence exist:(1)a sequence in is said to be -convergent to in , denoted by as , if there exists a delta sequence such that , , and as , in , for every ;(2)a sequence in is said to be -convergent to in , denoted by as , if there exists a such that , , and as in .

The following theorem is equivalent to the statement of -convergence.

Theorem 1. in if and only if there is and such that , and for each , as in .

For further discussion see [3, 1721].

3. Necessary and Sufficient Conditions

Denote by the space of Lebesgue complex-valued measurable functions of bounded support defined on and satisfying then being arbitrary but fixed.

By denote the space of test functions of bounded support defined on .

Let be the set of sequences such that [3, (2.6)–(2.8)]

The convolution product between two functions is defined by the integral equation where .

Now we construct the space of Boehmians.

We have the following definition.

Definition 2. Let and ; then we define the mapping as

Theorem 3. Let and ; then .

Proof. Let and . By using (10), Fubini’s theorem, and Jensen’s inequality we get where is a positive constant.

The proof is therefore completed.

Theorem 4. Let in and ; then as .

Proof of this theorem follows from Theorem 3.

Theorem 5. Let and ; then one has

Proof. Let . Using (10) and (9) we write

The substitution , , implies

This completes the proof of the theorem.

Theorem 6. Let and ; then

Theorem 7. Let , and ; then

Proof of Theorems 6 and 7 follows from simple integration. Detailed proof is thus avoided.

Theorem 8. Let and ; then as .

Proof. Let . Since is dense in , we can choose such that

From the analysis applied for proving Theorem 3 and by applying (17) we get Also, for each fixed define then and hence uniformly continuous on . Thus, there is such that whenever .

Moreover implies , .

Hence, by (8) and the fact that , by Jensen’s inequality, we have where is the Lebesgue measure of .

Hence, using (17), (18), and (20) we, for large values of , get Hence as .

The Boehmian space is constructed.

The sum and multiplication by a scalar of two Boehmians are naturally defined in the respective ways: being complex number.

The operations and the derivative are defined by

Between and the canonical embedding admits

The operation can be extended to by

By denote the corresponding Boehmian space obtained from , , and the product .

Theorem 9. Let and ; then

Proof. Let . By employing (2) for (9) we get

The substitution implies and .

Thus we get

This completes the proof of the theorem.

4. The Exponential Radon Transform of Boehmians

Definition 10. Let ; then we define its exponential Radon transform as the mapping in the space .

Definition 10 is well defined by Theorem 9.

To show that (29) is well defined, let and ; then Employing for (30) and using Theorem 9 imply that From (31) we see that in the sense of .

This completes the proof of the theorem.

Theorem 11. Let ; then .

Proof. Assume the requirements of the theorem are satisfied for some ; then there are and such that and . Therefore, we write

Thus we get that .

This completes the proof.

Theorem 12. defines a linear mapping from into .

The proof is straightforward.

Definition 13. Let be such that . Then we define the inverse transform of as for each .

Theorem 14. defines an isomorphism from onto .

Proof. Assume that in . Using (29) and Theorem 9 we get . Once again, Theorem 9 implies . Hence . Therefore, .

Now, let ; then , . Theorem 9 leads to . Hence is the Boehmian that satisfies .

This completes the proof of the theorem.

Theorem 15. Let and ; then

Proof. Applying Definition 10 for each and yields

By Theorem 9 we get

This completes the proof of the theorem.

Theorem 16. The mappings 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 Theorem 1 we can find and in such that and such that as for every .

Employing the continuity condition of transform implies as in the space .

Thus, in .

To prove the second part, let in as . Then, once again, by Theorem 1, and and as . Hence in as . That is, as . Using (33) we get

Now, we establish continuity of and with respect to convergence.

Let be such that as . Then, by Theorem 1 we can find that and such that and as . Employing (29) we get Hence, we have as in .

Therefore Hence, as .

Finally, let in as ; then we find such that and as for some .

Now, using (33), we obtain Theorem 9 implies Thus

From this we find that as .

This completes the proof of the theorem.

Theorem 17. The transform is consistent with .

Proof. For every , let be its representative; then , , . For all it is clear that is independent of the representative.

We have which is the representative of in the space .

Hence the proof is completed.

Theorem 18. The necessary and sufficient condition for 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 we get , .

Therefore, Theorem 9 yields where and , .

Thus , . Hence, The theorem is therefore completely proved.

Conflict of Interests

The authors declare that they have no conflict of interests regarding publication of this paper.

Acknowledgment

The second author gratefully acknowledges the partial support that this research received from the Universiti Putra Malaysia under GP-IBT Grant Scheme having Project no. GPIBT/2013/9420100.

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