# Note on Boehmians for Class of Optical Fresnel Wavelet Transforms

Authors: S. K. Q. Al-Omari; A. Kılıçman
Journal: Journal of Function Spaces and Applications
 (2012)

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## 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 toδ and Δ convergence are discussed in some details.


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## Body

## 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 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 [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
(1.1)Ωf(μ,λ)=1μ∫Rf(x)ψ*(x-λμ)dx,
where ψ(x) is named as the mother wavelet such that
(1.2)∫Rdxψ(x)=0,μ∈R+ and λ∈R 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)=12πiγ1∫Rexp[i2γ1(α1x12-2x1x2+α2x22)]f(x1)dx1.The parameters(α1,γ1,γ2,andα2) 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 parametersα1,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 [9]
(1.5)fw(x2)=12πiγ1∫RKλ,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 parametersα1,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 [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 Section2, 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

Let𝔼(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, 18–20]. Then, 𝔼(R) is a complete multinormed space with the set of norms as follows:
(2.1)ξk(ϕ)(x)=supx∈K|Dxkϕ(x)|,
where K run through compact subsets of R and ϕ∈𝔼(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,λ∈R,  μ∈R+ 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.
Letf,g∈𝔼′(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),whereTf*g(x2)=fw((f*g)(x))(x2).Proof.
Letf,g∈𝔼′(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.Letδ 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)),Nowe can easily deduce a corollary for the Lemma 2.1 as follows.Corollary 2.2.
Letf,g∈𝔼′(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 Lemma2.1.Theorem 2.3.
The distributional optical Fresnel-wavelet transformfw is linear.Proof.
It is obvious.Lemma 2.4.
Letf,g∈𝔼′(R),Tg(x2)=fw(g(x))(x2), Tf(x2)=fw(f(τ))(x2),   and then one has (1)
T(f*g)(k)(x2)=expi2γ1(-2x2λμ-α2x22)Tf(k)(x2)Tg(x2)(2)
T(f*g)(k)(x2)=expi2γ1(-2x2λμ-α2x22)Tf(x2)Tg(k)(x2).Proof.
It is a straightforward conclusion of the fact [20]. Consider
(2.10)(f*g)(k)=f(k)*g=f*g(k).


## 3. The Boehmian Space𝔹1

In this section, we assume that the reader is acquainted with the general construction of Boehmian spaces [6, 22–27]. Let 𝔻(R) be the Schwartz space of test functions of bounded support see [12, 20, 28]. The operation • between a distribution f∈𝔼′(R) and a test function ϕ∈𝔻(R) is defined by
(3.1)(f•g)ψ(x)=f(g⊙τxψ),
where τxψ(y)=ψ(x+y) and
(3.2)(g⊙τxψ)(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)dx≤M,∃M>0,(3.5)suppϕn⊂(-εn,εn)→0asn→∞.The set of all such sequences is denoted byΔ. To see the extension to certain integral transform, see [29–31].Lemma 3.1.
Givenϕ∈𝔻(R) and φ∈𝔼(R) then
(3.6)dkdxk(ϕ⊙τxφ)=ϕ⊙dkdxkτxφ,
for each k∈ℕ.Proof.
We prove the lemma by induction onk. Let k=1 and ϕ∈𝔻(R) with suppϕ⊂K, and then
(3.7)ddx(ϕ⊙τxφ)=limx→hϕ⊙τxφ-ϕ⊙τhφx-h=ϕ⊙limx→hτxφ-τhφx-h.
Hence (3.7) reduces to
(3.8)ddx(ϕ⊙τxφ)=ϕ⊙ddxτxφ.Next, assume that the lemma satisfies forkth derivatives, then certainly we get
(3.9)dk+1dxk+1(ϕ⊙τxφ)=ddx(dkdxk(ϕ⊙τxφ))=ϕ⊙dk+1dxk+1τxφ,
by (3.7). Hence the lemmais as follows.Lemma 3.2.
Letϕ∈𝔻(R) and φ∈𝔼(R), and then ϕ⊙τxφ∈𝔼(R).Proof.
LetK be a compact subset of R. Then using Lemma 3.1 we get
(3.10)ξk(ϕ⊙τxφ)≤supx∈K|dk+1dxk+1τxφ|.
The inequality (3.10) can be explicitly expressed as
(3.11)ξk(ϕ⊙τxφ)≤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.
Letf∈𝔼′(R) and  ϕ∈𝔻(R), and then  f•ϕ∈𝔼′(R).Proof.
In view of Lemma3.2 we get
(3.12)f⊙τxϕ∈𝔼(R).
Therefore, the righthand side of (3.1) is meaningful. To show that f•ϕ∈𝔼′(R) we are requested to show that f•ϕ is continuous and linear. To establish continuity, let (ψn)→0 in 𝔼(R), then from (3.11) we get
(3.13)ξk(f⊙τxψn)≤ξk(ψn)→0asn→∞.Hence we have(3.14)(f•ϕ)ψn(x)=f(ϕ⊙τxψn)→0
as n→∞. Linearity condition is obvious. Hence the lemma is completely proved.Lemma 3.4.
Letϕ1,ϕ2∈𝔻(R) and ψ∈𝔼(R) be given, and then
(3.15)(ϕ1*ϕ2)⊙τxψ=ϕ1•(ϕ2⊙τxψ)Proof.
It is a straightforward consequence of definitions and change of variables.Lemma 3.5.
Letϕ1,ϕ2∈𝔻(R) and  f∈𝔼′(R), and then
(3.16)f•(ϕ1*ϕ2)=(f•ϕ1)•ϕ2.Proof.
Using (3.1) and Lemma 3.4. we get
(3.17)(f•(ϕ1*ϕ2))ψ(x)=f((ϕ1*ϕ2)⊙τxψ)=f(ϕ1•(ϕ2⊙τxψ))=(f•ϕ1)•ϕ2.
Hence the lemma is as follows.Lemma 3.6.
Letf1,f2∈𝔼′(R) and  ϕ∈𝔻(R), and then one has (1)
αf•ϕ=α(f•ϕ),α∈ℂ(2)
(f1+f2)•ϕ=f1•ϕ+f2•ϕ.Proof.
It is a straightforward result of definitions.Lemma 3.7.
Letfn→f in 𝔼′(R) and ϕ∈𝔻(R) be given then
(3.18)fn•ϕ→f•ϕin𝔼′(R)
as n→∞.Proof.
By virtue of Lemma3.2, ϕ⊙τxψ∈𝔼(R). Hence, using (3.1) we get
(3.19)(fn•ϕ-f•ϕ)ψ(x)=((fn-f)•ϕ)ψ(x)=(fn-f)(ϕ⊙τxψ).
Allowingn→∞ completes the proof of the lemma.Lemma 3.8.
Letf∈𝔼′(R) and  (ϕn)∈Δ, and then  f•ϕ→f∈𝔼′(R).Proof.
Considering a compact subsetK of R and a sequence (ϕn)∈Δ such that suppϕn⊂(-εn,εn) for each n∈ℕ, we show that ϕ⊙τxψ→ψ(x) as n→∞ in the sense of 𝔼(R). By Lemma 3.1 we have
(3.20)|dkdxk(ϕ⊙τxψ-ψ(x))|≤∫-εnεn|dkdxk(ψ(x+y)-ψ(x))||ϕn(y)|dy,
and by applying (3.4) we get
(3.21)|dkdxk(ϕ⊙τxψ-ψ(x))|≤∫-εnεnM|dkdxk(ψ(x+y)-ψ(x))|dx.
Then the mean value theorem implies that(3.22)|dkdxk(ϕ⊙τxψ-ψ(x))|≤∫-εnεnM|y||dk+1dxk+1ψ(x+ξ)|dy
for some ξ∈(0,y). Let A=subs∈K|(dk+1/dsk)ψ(s)| then considering supremum over all x∈K with the fact that |y|≤εn yields
(3.23)ξk(ϕ⊙τxψ-ψ(x))≤AMεn.
Now allowingn→∞ in (3.23) yields ξk(ϕ⊙τxψ-ψ(x))→0. Hence we have established that
(3.24)ϕ⊙τxψ→ψ(x).
On using (3.24) can be observed as
(3.25)(f•ϕn)ψ(x)=f(ϕ⊙τxψ)→f(ψ)(x)asn→∞.
This implies thatf•ϕn→f as n→∞. The proof is therefore completed.Finally, by virtue of the above sequence of results (Lemma3.1–3.8), our desired Boehmian space 𝔹1 is well defined.


## 4. The Boehmian Space𝔹𝔽w

In this section we construct the space of all Fresnel-wavelet transforms of Boehmians from the space𝔹1 as follows.Let𝔽w(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 Tn→T in 𝔽w(R) if and only if there are fn,f∈𝔼′(R) such that fn→f in 𝔼′(R), where Tn=fwfn and T=fwf. Let T∈𝔽w(R) and ϕ∈𝔻(R) be given, and then define
(4.1)(T⋆ϕ)(x2)=ϕ⊙τx2T.Theorem 4.1.
Letf∈𝔼′(R) and  ϕ∈𝔻(R), and then  fw(f•ϕ)=T⋆ϕ, where T=fwf.Proof.
By the aid of (2.3) we write
(4.2)fw(f•ϕ)(x2)=〈f•ϕ,Kλ,μ,x2(x1)〉.
By using (3.1) we get
(4.3)fw(f•ϕ)(x2)=f(ϕ⊙τx2Kλ,μ,x2(x1))=ϕ(f⊙τx2Kλ,μ,x2(x1)).
That isfw(f•ϕ)(x2)=ϕ⊙τx2T. Employing (4.1) yields fw(f•ϕ)(x2)=(T⋆ϕ)(x2), where T=fwf. This proves the theorem.Lemma 4.2.
LetT∈𝔽w(R) and  ϕ∈𝔻(R), and then  T⋆ϕ∈𝔽w(R).Proof.
Letf∈𝔼′(R) be such that T=fwf, and that then Theorem 4.1 implies
(4.4)T⋆ϕ=fw(f•ϕ).
ThusT⋆ϕ∈𝔽w(R) by Lemma 3.3. Hence the lemma.Lemma 4.3.
LetT∈𝔽w(R) and  ϕ∈𝔻(R), and then fw-1(T⋆ϕ)=f•ϕ.Proof.
Theorem4.1 implies that fw-1(T⋆ϕ)=fw-1(fw(f•ϕ))=f•ϕ. This completes the proof of the lemma.Lemma 4.4.
LetT1,T2∈𝔽w(R) and  ϕ1,ϕ2∈𝔻(R), then (1)
(αT1+βT2)⋆ϕ=αT1⋆ϕ+βT2⋆ϕ,(2)
T⋆(ϕ1⋆ϕ2)=(T⋆ϕ1)⋆ϕ2.Proof.
It is obvious.Lemma 4.5.
(i) LetTn→T as n→∞ and ϕ∈𝔻(R), and then Tn⋆ϕ→T⋆ϕ as n→∞ in 𝔽w(R).
(ii) LetTn→T as n→∞ and (ϕn)∈Δ, and then Tn⋆ϕn→T as n→∞ in 𝔽w(R).Proof.
(i) LetTn,T∈𝔽w(R) then T=fwf  and Tn=fwfn  for some f,fn∈𝔼′(R). Hence, using  Theorem 4.1, we have
(4.5)Tn⋆ϕ=fwfn⋆ϕ=fw(fn•ϕ).
By Lemma3.7 we get
(4.6)Tn⋆ϕ→fw(f•ϕ).
Once again Theorem4.1 implies that
(4.7)Tn⋆ϕ→fwf⋆ϕ.
Thus we ahve
(4.8)Tn⋆ϕ→T⋆ϕ.
This proves part (i) of the Lemma.
(ii) can be proved similarly by using Lemma3.8 and Theorem 4.1. The space 𝔹𝔽w is therefore established.The sum of two Boehmians and multiplication by a scalar in𝔹𝔽w is defined in a natural way as follows:
(4.9)[fnϕn]+[gnψn]=[(fn⋆ψn)+(gn⋆ϕn)ϕn⋆ψn],α[fnϕn]=[αfnϕn],α∈ℂ.The operation⋆ and the differentiation are defined by
(4.10)[fnϕn]⋆[gnψn]=[fn⋆gnϕn⋆ψn],Dα[fnϕn]=[Dαfnϕn].


## 5. Optical Fresnel-Wavelet Transforms of Boehmians

In view of the analysis obtained in Sections4 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 transform𝔉w:𝔹1→𝔹𝔽w is well defined.Proof.
It is a straightforward.Lemma 5.3.
The optical Fresnel-wavelet transform𝔉w:𝔹1→𝔹𝔽w is linear.Proof.
It is straightforward by using Definition5.1.Lemma 5.4.
The optical Fresnel-wavelet transform𝔉w:𝔹1→𝔹𝔽w is an isomorphism.Proof.
Assume that𝔉w[fn/ϕn]=𝔉w[gn/ψn], and then using (5.1) and the concept of quotients we get Tfn⋆ψm=Tgm⋆ϕn, where Tfn=fwfn and Tgm=fwgm. Therefore, Theorem 4.1 implies that fw(fn•ψm)=fw(gm•ϕn). Properties of fw imply that fn•ψm=gm•ϕn. Therefore, [fn/ϕn]=[gn/ψn]. To establish fw is surjective, and let [Tfn/ϕn]∈𝔹Fw. Then Tfn⋆ϕm=Tfm⋆ϕn for every m,n∈ℕ. Hence fn,fm∈𝔼′(R) are such that Tfn=fwfn and Tfm=fwfm. Theorem 4.1 implies that fw(fn•ϕm)=fw(fm•ϕn). 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 ϕ∈𝔻(ℝ) then
(5.3)𝔉w-1([Tfnϕn]⋆ϕ)=[Tfn-1ϕn]•ϕ,𝔉w([fnϕn]•ϕ)=[Tfnϕn]⋆ϕ.Proof.
Applying Definition5.1 yields
(5.4)𝔉w-1([Tfnϕn]⋆ϕ)=𝔉w-1([Tfn⋆ϕϕn])=[fw-1(Tfn⋆ϕ)ϕn].
Using Lemma4.3 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:𝔹𝔽w→𝔹1 are continuous with respect to δ and Δ convergences.Proof.
First of all, we show that𝔉w:𝔹1→𝔹𝔽w and 𝔉w-1:𝔹𝔽w→𝔹1 are continuous with respect to δ convergence.
Letβn→δβ in 𝔹1 as n→∞, and then we show that 𝔉wβn→𝔉wβ as n→∞. By virtue of [25] we can find fn,k and fk in 𝔼′(R) such that
(5.6)βn=[fn,kϕk],β=[fkϕk]
such thatfn,k→fk as n→∞ for every k∈ℕ. Employing the continuity condition of the optical Fresnel-wavelet transform implies that Tfn,k→Tfk as n→∞ in the space 𝔽w(R). Thus, [Tfn,k/ϕk]→[Tfk/ϕk] as n→∞ in 𝔹𝔽w.
To prove the second part, letgn→δg in 𝔹𝔽w as n→∞. Then, once again, by [4], gn=[Tfn,k/ϕk] and g=[Tfk/ϕk] for some Tfn,k,Tfk∈𝔽w(R) and Tfn,k→Tfk as n→∞. Hence fw-1Tfn,k→fw-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 of𝔉w and 𝔉w-1 with respect to Δ convergence. Let βn→Δβ in 𝔹1 as n→∞. Then, there exist fn∈𝔼′(R) and ϕn∈Δ such that (βn-β)•ϕn=[(fn•ϕk)/ϕk] and fn→0 as n→∞. Employing Definition 5.1 we get
(5.7)𝔉w((βn-β)•ϕn)=[fw(fn•ϕk)ϕk].
Hence, from Lemma4.2 we have 𝔉w((βn-β)•ϕn)=[(Tfn⋆ϕk)/ϕk]=Tfn→0 as n→∞ in 𝔽w. Therefore consider
(5.8)𝔉w((βn-β)•ϕn)=(𝔉wβn-𝔉wβ)⋆ϕn→0asn→∞.
Hence,𝔉wβn→Δ𝔉wβ as n→∞. Finally, let gn→Δg in 𝔹𝔽w as n→∞, and then we find Tfk∈𝔽w(R) such that (gn-g)⋆ϕn=[(Tfk⋆ϕk)/ϕk] and Tfk→0 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(Tfk⋆ϕk)ϕk].
Lemma5.5 implies that
(5.10)𝔉w-1((gn-g)⋆ϕn)=[fn•ϕkϕk]=fn→0asn→∞in𝔼′(R).
Thus we have(5.11)𝔉w-1((gn-g)⋆ϕn)=(𝔉w-1gn-𝔉w-1g)•ϕn→0asn→∞.
From this we find that𝔉w-1gn→Δ𝔉w-1g as n→∞ in 𝔹1. This completes the proof of the theorem.



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*Source: 405368-2012-10-02.xml*