JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 405368 10.1155/2012/405368 405368 Research Article Note on Boehmians for Class of Optical Fresnel Wavelet Transforms Al-Omari S. K. Q. 1 Kılıçman A. 2 Ruiz Galan Manuel 1 Department of Applied Sciences, Faculty of Engineering Technology Al-Balqa Applied University, Amman 11134 Jordan bau.edu.jo 2 Department of Mathematics and Institute of Mathematical Research Universiti Putra Malaysia (UPM), Selangor, 43400 Serdang Malaysia upm.edu.my 2012 2 10 2012 2012 04 07 2012 27 08 2012 2012 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.

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, . 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 exp[(i/2c)(ax12+bx22)], for some constants a, b and c. 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 (1.1)Ωf(μ,λ)=1μRf(x)ψ*(x-λμ)dx, where ψ(x) is named as the mother wavelet such that (1.2)Rdxψ(x)=0,μR+ and λR are the transform dilate and translate of the wavelet ψ and ψ* being the complex conjugate of ψ. The optical diffraction transform is described by the Fresnel integration in [5, 9] as follows: (1.3)fw(x2)=12πiγ1Rexp[i2γ1(α1x12-2x1x2+α2x22)]f(x1)dx1.

The parameters (α1,γ1,γ2,and  α2) are elements of more ray transfer Matrix M describing optical systems, α1α2-γ1γ2=1. 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 α1,γ1,γ2, and α2 are written in the following matrix form: (1.4)(α1γ1γ2α2)=(cosθsinθ-sinθcosθ)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  (1.5)fw(x2)=12πiγ1RKλ,μ,x2(x1)f(x1)dx. with kernel (1.6)Kλ,μ,x2(x1)=exp(i2γ1(α1(x1-λ)2μ2-2x2(x1-λ)μ+α2x22)).

The parameters α1,γ1,γ2, and α2 appearing in (1.5) are elements of 2×2 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 𝔼(R) be the space of all test functions ϕ(x) of arbitrary support and 𝔼(R) be its dual of distributions of bounded support, see, for example [12, 1820]. Then, 𝔼(R) is a complete multinormed space with the set of norms as follows: (2.1)ξk(ϕ)(x)=supxK|Dxkϕ(x)|, where K run through compact subsets of R and ϕ𝔼(R). It is clear that the kernel function of the optical Fresnel-wavelet transform (2.2)Kλ,μ,x2(x1)=expi2γ1(α1(x1-λ)2μ2-2x2(x1-λ)μ+α2x22) for each x2,λR,  μR+ is an element of 𝔼(R). This describes the distributional optical Fresnel-wavelet transform of bounded support as an adjoint operator as follows: (2.3)fw(x2)=Tf(x2)=f(x1),expi2γ1(α1(x1-λ)2μ2-2x2(x1-λ)μ+α2x22). For convenience we sometimes write Tf(x2) instead of fw(x2). Moreover, from (2.3), we observe that Tf(x2) is an analytic function satisfying the expression as follows: (2.4)Dx2kTf(x2)=f(x1),Dx2kKλ,μ,x2(x1).

Further, Dx2kTf(x2) is well defined since Dx2kKλ,μ,x2(x1)𝔼(R),Kλ,μ,x2(x1) 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 f,g𝔼(R) and Tf=fw(f(x))(x2), Tg=fw(g(τ))(x2) be their respective optical Fresnel-wavelet transforms and λ2=2xτ for all x and τ, then (2.5)Tf*g(x2)=expi2γ1(-2x2λμ-α2x22)Tf(x2)Tg(x2),

where Tf*g(x2)=fw((f*g)(x))(x2).

Proof.

Let f,g𝔼(R),Tf=fw(f(x))(x2), and Tg=fw(g(τ))(x2), and then (2.6)Tf*g(x2)=(f*g)(x),expi2γ1(α1(x-λ)2μ2-2x2(x-λ)μ+α2x22)i.e.,=f(x),g(τ),expi2γ1(α1(x+τ-λ)2μ2-2x2(x+τ-λ)μ+α2x22). Hence, using properties of distributions and simple calculations we get (2.7)Tf*g(x2)=expi2γ1(-2x2λμ-α2x22)Tf(x2)Tg(x2).

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

Let δ be the dirac delta function. Then the Fresnel-wavelet transform of δ is described as follows: (2.8)Tδ=fw(δ(x))(x2)=exp(i2γ1(α1λ2μ2+2x2λμ+α2x22)),

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

Corollary 2.2.

Let f,g𝔼(R) and δ be the dirac delta function, and then (2.9)Tf*δ(x2)=expi2γ1(α14λ2+2xτμ2)Tf(x2),Tδ*g(x2)=expi2γ1(α14λ2+2xτμ2)Tg(x2).

Proof.

It is a straightforward result of Lemma 2.1.

Theorem 2.3.

The distributional optical Fresnel-wavelet transform fw is linear.

Proof.

It is obvious.

Lemma 2.4.

Let f,g𝔼(R),Tg(x2)=fw(g(x))(x2), Tf(x2)=fw(f(τ))(x2),   and then one has

T(f*g)(k)(x2)=expi2γ1(-2x2λμ-α2x22)Tf(k)(x2)Tg(x2)

T(f*g)(k)(x2)=expi2γ1(-2x2λμ-α2x22)Tf(x2)Tg(k)(x2).

Proof.

It is a straightforward conclusion of the fact . Consider (2.10)(f*g)(k)=f(k)*g=f*g(k).

3. The Boehmian Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M70"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝔹</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In this section, we assume that the reader is acquainted with the general construction of Boehmian spaces [6, 2227]. Let 𝔻(R) be the Schwartz space of test functions of bounded support see [12, 20, 28]. The operation between a distribution f𝔼(R) and a test function ϕ𝔻(R) is defined by (3.1)(fg)ψ(x)=f(gτxψ), where τxψ(y)=ψ(x+y) and (3.2)(gτxψ)(y)=Rg(y)τxψ(y)dy.

A sequence (ϕn) of functions in 𝔻(R) is said to be a delta sequence if it satisfies Conditions (3.3)–(3.5). Consider (3.3)Rϕn(x)dx=1(3.4)Rϕn(x)dxM,M>0,(3.5)suppϕn(-εn,εn)0asn.

The set of all such sequences is denoted by Δ. To see the extension to certain integral transform, see .

Lemma 3.1.

Given ϕ𝔻(R) and φ𝔼(R) then (3.6)dkdxk(ϕτxφ)=ϕdkdxkτxφ, for each k.

Proof.

We prove the lemma by induction on k. Let k=1 and ϕ𝔻(R) with suppϕK, and then (3.7)ddx(ϕτxφ)=limxhϕτxφ-ϕτhφx-h=ϕlimxhτxφ-τhφx-h. Hence (3.7) reduces to (3.8)ddx(ϕτxφ)=ϕddxτxφ.

Next, assume that the lemma satisfies for kth derivatives, then certainly we get (3.9)dk+1dxk+1(ϕτxφ)=ddx(dkdxk(ϕτxφ))=ϕdk+1dxk+1τxφ, by (3.7). Hence the lemmais as follows.

Lemma 3.2.

Let ϕ𝔻(R) and φ𝔼(R), and then ϕτxφ𝔼(R).

Proof.

Let K be a compact subset of R. Then using Lemma 3.1 we get (3.10)ξk(ϕτxφ)supxK|dk+1dxk+1τxφ|.

The inequality (3.10) can be explicitly expressed as (3.11)ξk(ϕτxφ)ξk(φ), where ξk is the norm in the topology equipped with 𝔼(R). Hence, the lemma follows from (3.11). This completes the proof.

Lemma 3.3.

Let f𝔼(R) and ϕ𝔻(R), and then fϕ𝔼(R).

Proof.

In view of Lemma 3.2 we get (3.12)fτxϕ𝔼(R).

Therefore, the righthand side of (3.1) is meaningful. To show that fϕ𝔼(R) we are requested to show that fϕ is continuous and linear. To establish continuity, let (ψn)0 in 𝔼(R), then from (3.11) we get (3.13)ξk(fτxψn)ξk(ψn)0as  n.

Hence we have (3.14)(fϕ)ψn(x)=f(ϕτxψn)0 as n. Linearity condition is obvious. Hence the lemma is completely proved.

Lemma 3.4.

Let ϕ1,ϕ2𝔻(R) and ψ𝔼(R) be given, and then (3.15)(ϕ1*ϕ2)τxψ=ϕ1(ϕ2τxψ)

Proof.

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

Lemma 3.5.

Let ϕ1,ϕ2𝔻(R) and f𝔼(R), and then (3.16)f(ϕ1*ϕ2)=(fϕ1)ϕ2.

Proof.

Using (3.1) and Lemma 3.4. we get (3.17)(f(ϕ1*ϕ2))ψ(x)=f((ϕ1*ϕ2)τxψ)=f(ϕ1(ϕ2τxψ))=(fϕ1)ϕ2. Hence the lemma is as follows.

Lemma 3.6.

Let f1,f2𝔼(R) and ϕ𝔻(R), and then one has

αfϕ=α(fϕ),α

(f1+f2)ϕ=f1ϕ+f2ϕ.

Proof.

It is a straightforward result of definitions.

Lemma 3.7.

Let fnf in 𝔼(R) and ϕ𝔻(R) be given then (3.18)fnϕfϕin𝔼(R) as n.

Proof.

By virtue of Lemma 3.2, ϕτxψ𝔼(R). Hence, using (3.1) we get (3.19)(fnϕ-fϕ)ψ(x)=((fn-f)ϕ)ψ(x)=(fn-f)(ϕτxψ).

Allowing n completes the proof of the lemma.

Lemma 3.8.

Let f𝔼(R) and (ϕn)Δ, and then fϕf𝔼(R).

Proof.

Considering a compact subset K of R and a sequence (ϕn)Δ such that suppϕn(-εn,εn) for each n, we show that ϕτxψψ(x) as n in the sense of 𝔼(R). By Lemma 3.1 we have (3.20)|dkdxk(ϕτxψ-ψ(x))|-εnεn|dkdxk(ψ(x+y)-ψ(x))||ϕn(y)|dy, and by applying (3.4) we get (3.21)|dkdxk(ϕτxψ-ψ(x))|-εnεnM|dkdxk(ψ(x+y)-ψ(x))|dx.

Then the mean value theorem implies that (3.22)|dkdxk(ϕτxψ-ψ(x))|-εnεnM|y||dk+1dxk+1ψ(x+ξ)|dy for some ξ(0,y). Let A=subsK|(dk+1/dsk)ψ(s)| then considering supremum over all xK with the fact that |y|εn yields (3.23)ξk(ϕτxψ-ψ(x))AMεn.

Now allowing n in (3.23) yields ξk(ϕτxψ-ψ(x))0. Hence we have established that (3.24)ϕτxψψ(x).

On using (3.24) can be observed as (3.25)(fϕn)ψ(x)=f(ϕτxψ)f(ψ)(x) as  n.

This implies that fϕnf as n. The proof is therefore completed.

Finally, by virtue of the above sequence of results (Lemma 3.13.8), our desired Boehmian space 𝔹1 is well defined.

4. The Boehmian Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M159"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝔹</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>𝔽</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In this section we construct the space of all Fresnel-wavelet transforms of Boehmians from the space 𝔹1 as follows.

Let 𝔽w(R) be the space of all analytic functions which are Fresnel-wavelet transforms of distributions in 𝔼(R). Then, we define convergence as follows. We say TnT in 𝔽w(R) if and only if there are fn,f𝔼(R) such that fnf in 𝔼(R), where Tn=fwfn and T=fwf. Let T𝔽w(R) and ϕ𝔻(R) be given, and then define (4.1)(Tϕ)(x2)=ϕτx2T.

Theorem 4.1.

Let f𝔼(R) and ϕ𝔻(R), and then fw(fϕ)=Tϕ, where T=fwf.

Proof.

By the aid of (2.3) we write (4.2)fw(fϕ)(x2)=fϕ,Kλ,μ,x2(x1).

By using (3.1) we get (4.3)fw(fϕ)(x2)=f(ϕτx2Kλ,μ,x2(x1))=ϕ(fτx2Kλ,μ,x2(x1)).

That is fw(fϕ)(x2)=ϕτx2T. Employing (4.1) yields fw(fϕ)(x2)=(Tϕ)(x2), where T=fwf. This proves the theorem.

Lemma 4.2.

Let T𝔽w(R) and ϕ𝔻(R), and then Tϕ𝔽w(R).

Proof.

Let f𝔼(R) be such that T=fwf, and that then Theorem 4.1 implies (4.4)Tϕ=fw(fϕ).

Thus Tϕ𝔽w(R) by Lemma 3.3. Hence the lemma.

Lemma 4.3.

Let T𝔽w(R) and ϕ𝔻(R), and then fw-1(Tϕ)=fϕ.

Proof.

Theorem 4.1 implies that fw-1(Tϕ)=fw-1(fw(fϕ))=fϕ. This completes the proof of the lemma.

Lemma 4.4.

Let T1,T2𝔽w(R) and ϕ1,ϕ2𝔻(R), then

(αT1+βT2)ϕ=αT1ϕ+βT2ϕ,

T(ϕ1ϕ2)=(Tϕ1)ϕ2.

Proof.

It is obvious.

Lemma 4.5.

(i) Let TnT as n and ϕ𝔻(R), and then TnϕTϕ as n in 𝔽w(R).

(ii) Let TnT as n and (ϕn)Δ, and then TnϕnT as n in 𝔽w(R).

Proof.

(i) Let Tn,T𝔽w(R) then T=fwf  and Tn=fwfn for some f,fn𝔼(R). Hence, using Theorem 4.1, we have (4.5)Tnϕ=fwfnϕ=fw(fnϕ).

By Lemma 3.7 we get (4.6)Tnϕfw(fϕ).

Once again Theorem 4.1 implies that (4.7)Tnϕfwfϕ. Thus we ahve (4.8)TnϕTϕ.

This proves part (i) of the Lemma.

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

The sum of two Boehmians and multiplication by a scalar in 𝔹𝔽w is defined in a natural way as follows: (4.9)[fnϕn]+[gnψn]=[(fnψn)+(gnϕn)ϕnψn],α[fnϕn]=[αfnϕn],      α.

The operation and the differentiation are defined by (4.10)[fnϕn][gnψn]=[fngnϕnψn],Dα[fnϕn]=[Dαfnϕn].

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 [fn/ϕn  ]𝔹1, and then (5.1)𝔉w-1[Tfnϕn]=[Tfn-1ϕn],𝔉w[fnϕn]=[Tfnϕn], for each (ϕn)Δ, where Tfn=fwfn.

Lemma 5.2.

The optical Fresnel-wavelet transform 𝔉w:𝔹1𝔹𝔽w is well defined.

Proof.

It is a straightforward.

Lemma 5.3.

The optical Fresnel-wavelet transform 𝔉w:𝔹1𝔹𝔽w is linear.

Proof.

It is straightforward by using Definition 5.1.

Lemma 5.4.

The optical Fresnel-wavelet transform 𝔉w:𝔹1𝔹𝔽w is an isomorphism.

Proof.

Assume that 𝔉w[fn/ϕn]=𝔉w[gn/ψn], and then using (5.1) and the concept of quotients we get Tfnψm=Tgmϕn, where Tfn=fwfn and Tgm=fwgm. Therefore, Theorem 4.1 implies that fw(fnψm)=fw(gmϕn). Properties of fw imply that fnψm=gmϕn. Therefore, [fn/ϕn]=[gn/ψn]. To establish fw is surjective, and let [Tfn/ϕn]𝔹Fw. Then Tfnϕm=Tfmϕn for every m,n. Hence fn,fm𝔼(R) are such that Tfn=fwfn and Tfm=fwfm. Theorem 4.1 implies that fw(fnϕm)=fw(fmϕn). Hence [fn/ϕn]𝔹1 is such that 𝔉w[fn/ϕn]=[Tfn/ϕn  ]. This completes the proof of the lemma.

Now, Let [fn/ϕn]𝔹1, and then we define the inverse optical Fresnel-wavelet transform of [fn/ϕn] as (5.2)𝔉w-1[fnϕn]=[Tfn-1ϕn], where Tfn-1=fw-1fn.

Lemma 5.5.

Let [Tfn/ϕn]𝔹𝔽w, Tfn=fwfn, and ϕ𝔻() then (5.3)𝔉w-1([Tfnϕn]ϕ)=[Tfn-1ϕn]ϕ,𝔉w([fnϕn]ϕ)=[Tfnϕn]ϕ.

Proof.

Applying Definition 5.1 yields (5.4)𝔉w-1([Tfnϕn]ϕ)=𝔉w-1([Tfnϕϕn])=[fw-1(Tfnϕ)ϕn].

Using Lemma 4.3 we obtain (5.5)𝔉w-1([Tfnϕn]ϕ)=[fw-1Tfnϕn]ϕ=[Tfn-1ϕn]ϕ.

This completes the proof of the Lemma.

Theorem 5.6.

𝔉 w : 𝔹 1 𝔹 𝔽 w and 𝔉w-1:𝔹𝔽w𝔹1 are continuous with respect to δ and Δ convergences.

Proof.

First of all, we show that 𝔉w:𝔹1𝔹𝔽w and 𝔉w-1:𝔹𝔽w𝔹1 are continuous with respect to δ convergence.

Let βnδβ in 𝔹1 as n, and then we show that 𝔉wβn𝔉wβ as n. By virtue of  we can find fn,k and fk in 𝔼(R) such that (5.6)βn=[fn,kϕk],β=[fkϕk]

such that fn,kfk as n for every k. Employing the continuity condition of the optical Fresnel-wavelet transform implies that Tfn,kTfk as n in the space 𝔽w(R). Thus, [Tfn,k/ϕk][Tfk/ϕk] as n in 𝔹𝔽w.

To prove the second part, let gnδg in 𝔹𝔽w as n. Then, once again, by , gn=[Tfn,k/ϕk] and g=[Tfk/ϕk] for some Tfn,k,Tfk𝔽w(R) and Tfn,kTfk as n. Hence fw-1Tfn,kfw-1Tfk in 𝔹1 as n. Or, [Tfn,k-1/ϕk][Tfk-1/ϕk] as n. Using Definition 5.1 we get 𝔉w-1[Tfn,k/ϕk]𝔉w-1[Tfk/ϕk] as n.

Now, we establish continuity of 𝔉w and 𝔉w-1 with respect to Δ convergence. Let βnΔβ in 𝔹1 as n. Then, there exist fn𝔼(R) and ϕnΔ such that (βn-β)ϕn=[(fnϕk)/ϕk] and fn0 as n. Employing Definition 5.1 we get (5.7)𝔉w((βn-β)ϕn)=[fw(fnϕk)ϕk].

Hence, from Lemma 4.2 we have 𝔉w((βn-β)ϕn)=[(Tfnϕk)/ϕk]=Tfn0 as n in 𝔽w. Therefore consider (5.8)𝔉w((βn-β)ϕn)=(𝔉wβn-𝔉wβ)ϕn0asn.

Hence,    𝔉wβnΔ𝔉wβ as n. Finally, let gnΔg in 𝔹𝔽w as n, and then we find Tfk𝔽w(R) such that (gn-g)ϕn=[(Tfkϕk)/ϕk] and Tfk0 as n for some (ϕn)Δ and Tfk=fwfn. Now, using Definition 5.1, we obtain that (5.9)𝔉w-1((gn-g)ϕn)=[fw-1(Tfkϕk)ϕk].

Lemma 5.5 implies that (5.10)𝔉w-1((gn-g)ϕn)=[fnϕkϕk]=fn0asnin𝔼(R).

Thus we have (5.11)𝔉w-1((gn-g)ϕn)=(𝔉w-1gn-𝔉w-1g)ϕn0asn.

From this we find that 𝔉w-1gnΔ𝔉w-1g as n in 𝔹1. 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.

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