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S. K. Q. Al-Omari, A. Kılıçman, "Note on Boehmians for Class of Optical Fresnel Wavelet Transforms", Journal of Function Spaces, vol. 2012, Article ID 405368, 14 pages, 2012. https://doi.org/10.1155/2012/405368
Note on Boehmians for Class of Optical Fresnel Wavelet Transforms
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.
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  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:
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  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 . 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:
Lemma 2.1. Let and , be their respective optical Fresnel-wavelet transforms and for all and , then
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 . 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
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
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 .
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.
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 .
Lemma 4.2. Let and , and then .
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.
(i) Let as and , and then as in .
(ii) Let as and , and then as in .
(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
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
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  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 , 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.
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.
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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.