Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 405368, 14 pages

doi:10.1155/2012/405368

Research Article

Note on Boehmians for Class of Optical Fresnel Wavelet Transforms

S. K. Q. Al-Omari¹ and A. Kılıçman²

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), Selangor, 43400 Serdang, Malaysia

Received 4 July 2012; Accepted 27 August 2012

Academic Editor: Manuel Ruiz Galan

Copyright © 2012 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 extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to δ and Δ convergence are discussed in some details.

1. Introduction

Optical integral transforms have been studied in several works, for example, [1–8]. However, is the Fresnel transform among all the great importance [5, 9] where for which the kernel takes the form of a complex exponential function , for some constants , and . The generalization of the Fresnel transform called the linear canonical transform was introduced in [10] and has recently attracted considerable attention in optics, see [4, 11]. One of the very well-known linear transform is the wavelet transform, see [12, 13] we have

where is named as the mother wavelet such that

and are the transform dilate and translate of the wavelet and being the complex conjugate of . The optical diffraction transform is described by the Fresnel integration in [5, 9] as follows:

The parameters are elements of more ray transfer Matrix describing optical systems, . For a details of Fresnel integrals, see [14, 15].

Note that many familiar transforms can be considered as special cases of the diffraction Fresnel transform. For example, if the parameters , and are written in the following matrix form:

then the diffraction Fresnel transform, the generalized Fresnel Transform becomes a fractional Fourier transform, see [11, 16, 17].

In the present work, we consider a combined optical transform of Fresnel and wavelet transforms, namely, the optical Fresnel-wavelet transform defined by [9]

with kernel

The parameters , and appearing in (1.5) are elements of matrix with unit determinant.

As the general single-mode squeezing operator of the generalized Fresnel transform is in wave optics, further its applications are having a faithful representation in the optical Fresnel-wavelet transform, see [9]. Therefore the combined optical Fresnel-wavelet transform can be more conveniently studied by the general single-mode squeezed operation.

However, our discussion is somewhat different and making more interesting. Since the theory of the optical Fresnel-wavelet transform of generalized functions has not been reported in the literature. Thus, we extend the optical Fresnel-wavelet transform to a specific space of generalized functions, namely, known as Boehmian space. In Section 2, we observe that the kernel function of the Fresnel-wavelet transform is a smooth function, and therefore the optical Fresnel-wavelet transform is defined as an adjoint operator in the space of distributions. In a concrete way, Section 3 builds an appropriate space of Boehmians, whereas Section 4 constructs a new space of all images of Boehmians from Section 3. In Section 5, we define the optical Fresnel-wavelet transform of a Boehmian and study some of its general properties.

2. Optical Fresnel-Wavelet Transforms of Distributions

Let be the space of all test functions of arbitrary support and be its dual of distributions of bounded support, see, for example [12, 18–20]. Then, is a complete multinormed space with the set of norms as follows:

where run through compact subsets of and . It is clear that the kernel function of the optical Fresnel-wavelet transform

for each ,   is an element of . This describes the distributional optical Fresnel-wavelet transform of bounded support as an adjoint operator as follows:

For convenience we sometimes write instead of . Moreover, from (2.3), we observe that is an analytic function satisfying the expression as follows:

Further, is well defined since has its usual meaning where denoted by to be the usual convolution product [18, 20, 21]. Then we have the following lemma.

Lemma 2.1. Let and , be their respective optical Fresnel-wavelet transforms and for all and , then

where .

Proof. Let , and , and then

Hence, using properties of distributions and simple calculations we get

The above theorem is known as the convolution theorem of the Fresnel-wavelet transform.

Let be the dirac delta function. Then the Fresnel-wavelet transform of is described as follows:

Now can easily deduce a corollary for the Lemma 2.1 as follows.

Corollary 2.2. Let and be the dirac delta function, and then

Proof. It is a straightforward result of Lemma 2.1.

Theorem 2.3. The distributional optical Fresnel-wavelet transform is linear.

Proof. It is obvious.

Lemma 2.4. Let , ,   and then one has

(1)(2).

Proof. It is a straightforward conclusion of the fact [20]. Consider

3. The Boehmian Space

In this section, we assume that the reader is acquainted with the general construction of Boehmian spaces [6, 22–27]. Let be the Schwartz space of test functions of bounded support see [12, 20, 28]. The operation between a distribution and a test function is defined by

where and

A sequence of functions in is said to be a delta sequence if it satisfies Conditions (3.3)–(3.5). Consider

The set of all such sequences is denoted by . To see the extension to certain integral transform, see [29–31].

Lemma 3.1. Given and then

for each .

Proof. We prove the lemma by induction on . Let and with , and then

Hence (3.7) reduces to

Next, assume that the lemma satisfies for th derivatives, then certainly we get

by (3.7). Hence the lemmais as follows.

Lemma 3.2. Let and , and then .

Proof. Let be a compact subset of . Then using Lemma 3.1 we get

The inequality (3.10) can be explicitly expressed as

where is the norm in the topology equipped with . Hence, the lemma follows from (3.11). This completes the proof.

Lemma 3.3. Let and , and then .

Proof. In view of Lemma 3.2 we get

Therefore, the righthand side of (3.1) is meaningful. To show that we are requested to show that is continuous and linear. To establish continuity, let in , then from (3.11) we get

Hence we have

as . Linearity condition is obvious. Hence the lemma is completely proved.

Lemma 3.4. Let and be given, and then

Proof. It is a straightforward consequence of definitions and change of variables.

Lemma 3.5. Let and , and then

Proof. Using (3.1) and Lemma 3.4. we get

Hence the lemma is as follows.

Lemma 3.6. Let and , and then one has

(1)(2).

Proof. It is a straightforward result of definitions.

Lemma 3.7. Let in and be given then

as .

Proof. By virtue of Lemma 3.2, . Hence, using (3.1) we get

Allowing completes the proof of the lemma.

Lemma 3.8. Let and , and then .

Proof. Considering a compact subset of and a sequence such that for each , we show that as in the sense of . By Lemma 3.1 we have

and by applying (3.4) we get

Then the mean value theorem implies that

for some . Let then considering supremum over all with the fact that yields

Now allowing in (3.23) yields . Hence we have established that

On using (3.24) can be observed as

This implies that as . The proof is therefore completed.

Finally, by virtue of the above sequence of results (Lemma 3.1–3.8), our desired Boehmian space is well defined.

4. The Boehmian Space

In this section we construct the space of all Fresnel-wavelet transforms of Boehmians from the space as follows.

Let be the space of all analytic functions which are Fresnel-wavelet transforms of distributions in . Then, we define convergence as follows. We say in if and only if there are such that in , where and . Let and be given, and then define

Theorem 4.1. Let and , and then , where .

Proof. By the aid of (2.3) we write

By using (3.1) we get

That is . Employing (4.1) yields , where . This proves the theorem.

Lemma 4.2. Let and , and then .

Proof. Let be such that , and that then Theorem 4.1 implies

Thus by Lemma 3.3. Hence the lemma.

Lemma 4.3. Let and , and then .

Proof. Theorem 4.1 implies that . This completes the proof of the lemma.

Lemma 4.4. Let and , then

(1), (2).

Proof. It is obvious.

Lemma 4.5. (i) Let as and , and then as in .

(ii) Let as and , and then as in .

Proof. (i) Let then   and for some . Hence, using Theorem 4.1, we have

By Lemma 3.7 we get

Once again Theorem 4.1 implies that

Thus we ahve

This proves part (i) of the Lemma.

(ii) can be proved similarly by using Lemma 3.8 and Theorem 4.1. The space is therefore established.

The sum of two Boehmians and multiplication by a scalar in is defined in a natural way as follows:

The operation and the differentiation are defined by

5. Optical Fresnel-Wavelet Transforms of Boehmians

In view of the analysis obtained in Sections 4 and 5 and Theorem 4.1 we are led to state the following definition.

Definition 5.1. Let , and then

for each , where .

Lemma 5.2. The optical Fresnel-wavelet transform is well defined.

Proof. It is a straightforward.

Lemma 5.3. The optical Fresnel-wavelet transform is linear.

Proof. It is straightforward by using Definition 5.1.

Lemma 5.4. The optical Fresnel-wavelet transform is an isomorphism.

Proof. Assume that , and then using (5.1) and the concept of quotients we get , where and . Therefore, Theorem 4.1 implies that . Properties of imply that . Therefore, . To establish is surjective, and let . Then for every . Hence are such that and . Theorem 4.1 implies that . Hence is such that . This completes the proof of the lemma.

Now, Let , and then we define the inverse optical Fresnel-wavelet transform of as

where .

Lemma 5.5. Let , , and then

Proof. Applying Definition 5.1 yields

Using Lemma 4.3 we obtain

This completes the proof of the Lemma.

Theorem 5.6. 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 show that as . By virtue of [25] we can find and in such that

such that as for every . Employing the continuity condition of the optical Fresnel-wavelet transform implies that as in the space . Thus, as in .

To prove the second part, let in as . Then, once again, by [4], and for some and as . Hence in as . Or, as . Using Definition 5.1 we get as .

Now, we establish continuity of and with respect to convergence. Let in as . Then, there exist and such that and as . Employing Definition 5.1 we get

Hence, from Lemma 4.2 we have as in . Therefore consider

Hence, as . Finally, let in as , and then we find such that and as for some and . Now, using Definition 5.1, we obtain that

Lemma 5.5 implies that

Thus we have

From this we find that as in . This completes the proof of the theorem.

Acknowledgment

The authors would like to express their sincere thanks and gratitude to the reviewer(s) for their valuable comments and suggestions for the improvement of this paper.

References

P. K. Banerji, S. K. Al-Omari, and L. Debnath, “Tempered distributional Fourier sine (cosine) transform,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 759–768, 2006.

J. W. Goodman, Introductionto Fourier Optics, Mc Graw-Hill, New York, NY, USA, 1972.

H.-Y. Fan and H. L. Lu, “Wave-function transformations by general SU(1, 1) single-mode squeezing and analogy to Fresnel transformations in wave optics,” Optics Communications, vol. 258, no. 1, pp. 51–58, 2006.

H.-Y. Fan and L. Y. Hu, “Optical Fresnel transformation and quantum tomography,” Optics Communications, vol. 282, no. 18, pp. 3734–3736, 2009.

H. Kogelink, “On the propagation of Gaussian beams of light through Lenslike media including those with a loss or gain variation,” Applied Optics, vol. 4, pp. 121562–121569, 1965.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” Journal of the Optical Society of America A, vol. 10, no. 10, pp. 2181–2186, 1993.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA Journal of Applied Mathematics, vol. 39, no. 2, pp. 159–175, 1987.

P. Mikusiński, “Fourier transform for integrable Boehmians,” The Rocky Mountain Journal of Mathematics, vol. 17, no. 3, pp. 577–582, 1987.

H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” Journal of the Optical Society of America A, vol. 10, no. 12, pp. 2522–2531, 1993.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” Journal of Mathematical Physics, vol. 12, 9 pages, 1971.

S. K. Q. Al-Omari and A. Kılıçman, “On diffraction Fresnel transforms for Boehmians,” Abstract and Applied Analysis, vol. 2011, Article ID 712746, 11 pages, 2011.

M. Holschneider, Wavelets, an Analysis Tool, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 1995.

M. Holschneider, “Wavelet analysis on the circle,” Journal of Mathematical Physics, vol. 31, no. 1, pp. 39–44, 1990.

A. Kılıçman, “On the Fresnel sine integral and the convolution,” International Journal of Mathematics and Mathematical Sciences, no. 37, pp. 2327–2333, 2003.

A. Kılıçman and B. Fisher, “On the Fresnel integrals and the convolution,” International Journal of Mathematics and Mathematical Sciences, no. 41, pp. 2635–2643, 2003.

L. M. Bernardo and O. D. Soaares, “Fractional Fourier transform and optical systems,” Optics Communications, vol. 110, pp. 517–522, 1994.

L. Mertz, Transformations in Optics, Wiley, New York, NY, USA, 1965.

A. Kılıçman, “A comparison on the commutative neutrix convolution of distributions and the exchange formula,” Czechoslovak Mathematical Journal, vol. 51(126), no. 3, pp. 463–471, 2001.

A. Kılıçman, “On the commutative neutrix product of distributions,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 8, pp. 753–762, 1999.

R. Roopkumar, “Mellin transform for Boehmians,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 4, no. 1, pp. 75–96, 2009.

H. Eltayeb, A. Kılıçman, and B. Fisher, “A new integral transform and associated distributions,” Integral Transforms and Special Functions, vol. 21, no. 5-6, pp. 367–379, 2010.

S. K. Q. Al-Omari, D. Loonker, P. K. Banerji, and S. L. Kalla, “Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces,” Integral Transforms and Special Functions, vol. 19, no. 5-6, pp. 453–462, 2008.

T. K. Boehme, “The support of Mikusiński operators,” Transactions of the American Mathematical Society, vol. 176, pp. 319–334, 1973.

D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” Journal of the Optical Society of America A, vol. 10, no. 9, pp. 1875–1881, 1993.

P. Mikusiński, “Tempered Boehmians and ultradistributions,” Proceedings of the American Mathematical Society, vol. 123, no. 3, pp. 813–817, 1995.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” Journal of the Institute of Mathematics and its Applications, vol. 25, no. 3, pp. 241–265, 1980.

R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science, Amsterdam, The Netherlands, 1997.

B. Fisher and A. Kılıçman, “A commutative neutrix product of ultradistributions,” Integral Transforms and Special Functions, vol. 4, no. 1-2, pp. 77–82, 1996.

S. Al-Omari, “The generalized stieltjes and Fourier transforms of certain spaces of generalized functions,” The Jordanian Journal of Mathematics and Statistics, vol. 2, no. 2, pp. 55–66, 2009.

S. K. Q. Al-Omari, “On the distributional Mellin transformation and its extension to Boehmian spaces,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 17-20, pp. 801–810, 2011.

S. K. Q. Al-Omari, “A Mellin transform for a space of Lebesgue integrable Boehmians,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 29-32, pp. 1597–1606, 2011.