Note on Boehmians for Class of Optical Fresnel Wavelet Transforms

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 exp[(𝑖/2𝑐)(𝑎𝑥²₁+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 Ω_𝑓=1𝜇,𝜆√𝜇_𝐑𝑓(𝑥)𝜓^∗𝑥−𝜆𝜇𝑑𝑥, (1.1) where 𝜓(𝑥) is named as the mother wavelet such that 𝑑𝑥𝜓(_𝐑𝑥)=0, (1.2)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: 𝑓_𝑤𝑥₂=12𝜋𝑖𝛾_1𝐑𝑖exp2𝛾₁𝛼₁𝑥²₁−2𝑥₁𝑥₂+𝛼₂𝑥²₂𝑓𝑥₁𝑑𝑥₁. (1.3)

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 parameters 𝛼₁,𝛾₁,𝛾₂, and 𝛼₂ are written in the following matrix form: 𝛼₁𝛾₁𝛾₂𝛼₂=cos𝜃sin𝜃−sin𝜃cos𝜃 (1.4)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] 𝑓_𝑤𝑥₂=12𝜋𝑖𝛾_1𝐑𝐾_{𝜆,𝜇,𝑥2}𝑥₁𝑓𝑥₁𝑑𝑥. (1.5) with kernel 𝐾_{𝜆,𝜇,𝑥2}𝑥₁𝑖=exp2𝛾₁𝛼₁𝑥₁−𝜆²𝜇²−2𝑥₂𝑥₁−𝜆𝜇+𝛼₂𝑥²₂. (1.6)

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 [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 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, 18–20]. Then, 𝔼(𝐑) is a complete multinormed space with the set of norms as follows: 𝜉_𝑘𝜙(𝑥)=sup_𝑥∈𝐊𝐷^𝑘_𝑥𝜙(𝑥), (2.1) 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 𝐾_{𝜆,𝜇,𝑥2}𝑥₁𝑖=exp2𝛾₁𝛼₁𝑥₁−𝜆²𝜇²−2𝑥₂𝑥₁−𝜆𝜇+𝛼₂𝑥²₂ (2.2) 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: 𝑓_𝑤𝑥₂=𝑇_𝑓𝑥₂=𝑓𝑥₁𝑖,exp2𝛾₁𝛼₁𝑥₁−𝜆²𝜇²−2𝑥₂𝑥₁−𝜆𝜇+𝛼₂𝑥²₂. (2.3) For convenience we sometimes write 𝑇_𝑓(𝑥₂) instead of 𝑓_𝑤(𝑥₂). Moreover, from (2.3), we observe that 𝑇_𝑓(𝑥₂) is an analytic function satisfying the expression as follows: 𝐷^𝑘_𝑥2𝑇_𝑓𝑥₂=𝑓𝑥₁,𝐷^𝑘_𝑥2𝐾_{𝜆,𝜇,𝑥2}𝑥₁. (2.4)

Further, 𝐷^𝑘_𝑥2𝑇_𝑓(𝑥₂) is well defined since 𝐷^𝑘_𝑥2𝐾_{𝜆,𝜇,𝑥2}(𝑥₁)∈𝔼(𝐑),𝐾_{𝜆,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,𝑔∈𝔼^′(𝐑) and 𝑇_𝑓=𝑓_𝑤(𝑓(𝑥))(𝑥₂), 𝑇_𝑔=𝑓_𝑤(𝑔(𝜏))(𝑥₂) be their respective optical Fresnel-wavelet transforms and 𝜆²=2𝑥𝜏 for all 𝑥 and 𝜏, then 𝑇_𝑓∗𝑔𝑥₂𝑖=exp2𝛾₁−2𝑥₂𝜆𝜇−𝛼₂𝑥²₂𝑇_𝑓𝑥₂𝑇_𝑔𝑥₂, (2.5)

where 𝑇_𝑓∗𝑔(𝑥₂)=𝑓_𝑤((𝑓∗𝑔)(𝑥))(𝑥₂).

Proof. Let 𝑓,𝑔∈𝔼^′(𝐑),𝑇_𝑓=𝑓_𝑤(𝑓(𝑥))(𝑥₂), and 𝑇_𝑔=𝑓_𝑤(𝑔(𝜏))(𝑥₂), and then 𝑇_𝑓∗𝑔𝑥₂=(𝑥)𝑖𝑓∗𝑔,exp2𝛾₁𝛼₁(𝑥−𝜆)²𝜇²−2𝑥₂(𝑥−𝜆)𝜇+𝛼₂𝑥²₂𝑓(𝑥),𝑔(𝜏)𝑖i.e.,=,exp2𝛾₁𝛼₁(𝑥+𝜏−𝜆)²𝜇²−2𝑥₂(𝑥+𝜏−𝜆)𝜇+𝛼₂𝑥²₂. (2.6) Hence, using properties of distributions and simple calculations we get 𝑇_𝑓∗𝑔𝑥₂𝑖=exp2𝛾₁−2𝑥₂𝜆𝜇−𝛼₂𝑥²₂𝑇_𝑓𝑥₂𝑇_𝑔𝑥₂. (2.7)
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: 𝑇_𝛿=𝑓_𝑤(𝛿(𝑥))𝑥₂𝑖=exp2𝛾₁𝛼₁𝜆²𝜇²+2𝑥₂𝜆𝜇+𝛼₂𝑥²₂, (2.8)

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

Corollary 2.2. Let 𝑓,𝑔∈𝔼^′(𝐑) and 𝛿 be the dirac delta function, and then 𝑇_f∗𝛿𝑥₂𝑖=exp2𝛾₁𝛼₁4𝜆²+2𝑥𝜏𝜇²𝑇_𝑓𝑥₂,𝑇_𝛿∗𝑔𝑥₂𝑖=exp2𝛾₁𝛼₁4𝜆²+2𝑥𝜏𝜇²𝑇_𝑔𝑥₂. (2.9)

Proof. It is a straightforward result of Lemma 2.1.

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

Proof. It is obvious.

Lemma 2.4. Let 𝑓,𝑔∈𝔼^′(𝐑),𝑇_𝑔(𝑥₂)=𝑓_𝑤(𝑔(𝑥))(𝑥₂), 𝑇_𝑓(𝑥₂)=𝑓_𝑤(𝑓(𝜏))(𝑥₂),   and then one has (1)𝑇_{(𝑓∗𝑔)}^(𝑘)(𝑥₂𝑖)=exp2𝛾₁−2𝑥₂𝜆𝜇−𝛼₂𝑥²₂𝑇_𝑓^(𝑘)(𝑥₂)𝑇_𝑔(𝑥₂)(2)𝑇_{(𝑓∗𝑔)}^(𝑘)(𝑥₂𝑖)=exp2𝛾₁−2𝑥₂𝜆𝜇−𝛼₂𝑥²₂𝑇_𝑓(𝑥₂)𝑇_𝑔(𝑘)(𝑥₂).

Proof. It is a straightforward conclusion of the fact [20]. Consider 𝑓∗𝑔^(𝑘)=𝑓^(𝑘)∗𝑔=𝑓∗𝑔^(𝑘). (2.10)

3. The Boehmian Space 𝔹₁

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 𝑓•𝑔𝜓(𝑥)=𝑓𝑔⊙𝜏_𝑥𝜓, (3.1) where 𝜏_𝑥𝜓(𝑦)=𝜓(𝑥+𝑦) and 𝑔⊙𝜏_𝑥𝜓𝑦=_𝐑𝑔𝑦𝜏_𝑥𝜓𝑦𝑑𝑦. (3.2)

A sequence (𝜙_𝑛) of functions in 𝔻(𝐑) is said to be a delta sequence if it satisfies Conditions (3.3)–(3.5). Consider _𝐑𝜙_𝑛(𝑥)𝑑𝑥=1_𝐑𝜙_𝑛(𝑥)𝑑𝑥≤𝑀,∃𝑀>0,supp𝜙_𝑛⊂−𝜀_𝑛,𝜀_𝑛→0as𝑛→∞. (3.3)(3.4)(3.5)

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

Lemma 3.1. Given 𝜙∈𝔻(𝐑) and 𝜑∈𝔼(𝐑) then 𝑑^𝑘𝑑𝑥^𝑘𝜙⊙𝜏_𝑥𝜑𝑑=𝜙⊙^𝑘𝑑𝑥^𝑘𝜏_𝑥𝜑, (3.6) for each 𝑘∈ℕ.

Proof. We prove the lemma by induction on 𝑘. Let 𝑘=1 and 𝜙∈𝔻(𝐑) with supp𝜙⊂𝐾, and then 𝑑𝑑𝑥𝜙⊙𝜏_𝑥𝜑=lim_𝑥→ℎ𝜙⊙𝜏_𝑥𝜑−𝜙⊙𝜏_ℎ𝜑𝑥−ℎ=𝜙⊙lim_𝑥→ℎ𝜏_𝑥𝜑−𝜏_ℎ𝜑.𝑥−ℎ (3.7) Hence (3.7) reduces to 𝑑𝑑𝑥𝜙⊙𝜏_𝑥𝜑𝑑=𝜙⊙𝜏𝑑𝑥^𝑥𝜑. (3.8)

Next, assume that the lemma satisfies for 𝑘th derivatives, then certainly we get 𝑑^𝑘+1𝑑𝑥^𝑘+1𝜙⊙𝜏_𝑥𝜑=𝑑𝑑𝑑𝑥𝑑𝑥^𝑘𝑘𝜙⊙𝜏_𝑥𝜑𝑑=𝜙⊙^𝑘+1𝑑𝑥^𝑘+1𝜏_𝑥𝜑, (3.9) by (3.7). Hence the lemmais as follows.

Lemma 3.2. Let 𝜙∈𝔻(𝐑) and 𝜑∈𝔼(𝐑), and then 𝜙⊙𝜏_𝑥𝜑∈𝔼(𝐑).

Proof. Let 𝐾 be a compact subset of 𝐑. Then using Lemma 3.1 we get 𝜉_𝑘𝜙⊙𝜏_𝑥𝜑≤sup_𝑥∈𝐾𝑑^𝑘+1𝑑𝑥^𝑘+1𝜏_𝑥𝜑. (3.10)
The inequality (3.10) can be explicitly expressed as 𝜉_𝑘𝜙⊙𝜏_𝑥𝜑≤𝜉_𝑘𝜑, (3.11) where 𝜉_𝑘 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 𝑓∈𝔼^′(𝐑) and 𝜙∈𝔻(𝐑), and then 𝑓•𝜙∈𝔼^′(𝐑).

Proof. In view of Lemma 3.2 we get 𝑓⊙𝜏_𝑥𝜙∈𝔼(𝐑). (3.12)
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 (𝜓_𝑛)→0 in 𝔼(𝐑), then from (3.11) we get 𝜉_𝑘𝑓⊙𝜏_𝑥𝜓_𝑛≤𝜉_𝑘𝜓_𝑛→0as𝑛→∞. (3.13)

Hence we have 𝑓•𝜙𝜓_𝑛(𝑥)=𝑓𝜙⊙𝜏_𝑥𝜓_𝑛→0 (3.14) as 𝑛→∞. Linearity condition is obvious. Hence the lemma is completely proved.

Lemma 3.4. Let 𝜙₁,𝜙₂∈𝔻(𝐑) and 𝜓∈𝔼(𝐑) be given, and then 𝜙₁∗𝜙₂⊙𝜏_𝑥𝜓=𝜙₁•𝜙₂⊙𝜏_𝑥𝜓 (3.15)

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

Lemma 3.5. Let 𝜙₁,𝜙₂∈𝔻(𝐑) and 𝑓∈𝔼^′(𝐑), and then 𝑓•𝜙₁∗𝜙₂=𝑓•𝜙₁•𝜙₂. (3.16)

Proof. Using (3.1) and Lemma 3.4. we get 𝑓•𝜙₁∗𝜙₂𝜓(𝑥)=𝑓𝜙₁∗𝜙₂⊙𝜏_𝑥𝜓=𝑓𝜙₁•𝜙₂⊙𝜏_𝑥𝜓=𝑓•𝜙₁•𝜙₂. (3.17) Hence the lemma is as follows.

Lemma 3.6. Let 𝑓₁,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. Let 𝑓_𝑛→𝑓 in 𝔼^′(𝐑) and 𝜙∈𝔻(𝐑) be given then 𝑓_𝑛•𝜙→𝑓•𝜙in𝔼^′(𝐑) (3.18) as 𝑛→∞.

Proof. By virtue of Lemma 3.2, 𝜙⊙𝜏_𝑥𝜓∈𝔼(𝐑). Hence, using (3.1) we get 𝑓_𝑛•𝜙−𝑓•𝜙𝜓(𝑥)=𝑓_𝑛𝜓(𝑥)=𝑓−𝑓•𝜙_𝑛−𝑓𝜙⊙𝜏_𝑥𝜓. (3.19)
Allowing 𝑛→∞ completes the proof of the lemma.

Lemma 3.8. Let 𝑓∈𝔼^′(𝐑) and (𝜙_𝑛)∈Δ, and then 𝑓•𝜙→𝑓∈𝔼^′(𝐑).

Proof. Considering a compact subset 𝐾 of 𝐑 and a sequence (𝜙_𝑛)∈Δ such that supp𝜙_𝑛⊂(−𝜀_𝑛,𝜀_𝑛) for each 𝑛∈ℕ, we show that 𝜙⊙𝜏_𝑥𝜓→𝜓(𝑥) as 𝑛→∞ in the sense of 𝔼(𝐑). By Lemma 3.1 we have 𝑑^𝑘𝑑𝑥^𝑘𝜙⊙𝜏_𝑥𝜓−𝜓(𝑥)≤^𝜀𝑛_−𝜀𝑛𝑑^𝑘𝑑𝑥^𝑘𝜓𝑥+𝑦−𝜓(𝑥)𝜙_𝑛𝑦𝑑𝑦, (3.20) and by applying (3.4) we get 𝑑^𝑘𝑑𝑥^𝑘𝜙⊙𝜏_𝑥𝜓−𝜓(𝑥)≤^𝜀𝑛_−𝜀𝑛𝑀𝑑^𝑘𝑑𝑥^𝑘𝜓𝑥+𝑦−𝜓(𝑥)𝑑𝑥. (3.21)
Then the mean value theorem implies that 𝑑^𝑘𝑑𝑥^𝑘𝜙⊙𝜏_𝑥𝜓−𝜓(𝑥)≤^𝜀𝑛_−𝜀𝑛𝑀𝑦𝑑^𝑘+1𝑑𝑥^𝑘+1𝜓(𝑥+𝜉)𝑑𝑦 (3.22) for some 𝜉∈(0,𝑦). Let 𝐴=sub_𝑠∈𝐾|(𝑑^𝑘+1/𝑑𝑠^𝑘)𝜓(𝑠)| then considering supremum over all 𝑥∈𝐾 with the fact that |𝑦|≤𝜀_𝑛 yields 𝜉_𝑘𝜙⊙𝜏_𝑥𝜓−𝜓(𝑥)≤𝐴𝑀𝜀_𝑛. (3.23)
Now allowing 𝑛→∞ in (3.23) yields 𝜉_𝑘(𝜙⊙𝜏_𝑥𝜓−𝜓(𝑥))→0. Hence we have established that 𝜙⊙𝜏_𝑥𝜓→𝜓(𝑥). (3.24)
On using (3.24) can be observed as 𝑓•𝜙_𝑛𝜓(𝑥)=𝑓𝜙⊙𝜏_𝑥𝜓→𝑓𝜓(𝑥)as𝑛→∞. (3.25)
This implies that 𝑓•𝜙_𝑛→𝑓 as 𝑛→∞. The proof is therefore completed.

Finally, by virtue of the above sequence of results (Lemma 3.1–3.8), our desired Boehmian space 𝔹₁ is well defined.

4. The Boehmian Space 𝔹_𝔽𝑤

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

Let 𝔽_𝑤(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 𝑇_𝑛→𝑇 in 𝔽_𝑤(𝐑) if and only if there are 𝑓_𝑛,𝑓∈𝔼^′(𝐑) such that 𝑓_𝑛→𝑓 in 𝔼^′(𝐑), where 𝑇_𝑛=𝑓_𝑤𝑓_𝑛 and 𝑇=𝑓_𝑤𝑓. Let 𝑇∈𝔽_𝑤(𝐑) and 𝜙∈𝔻(𝐑) be given, and then define 𝑇⋆𝜙𝑥₂=𝜙⊙𝜏_𝑥2𝑇. (4.1)

Theorem 4.1. Let 𝑓∈𝔼^′(𝐑) and 𝜙∈𝔻(𝐑), and then 𝑓_𝑤(𝑓•𝜙)=𝑇⋆𝜙, where 𝑇=𝑓_𝑤𝑓.

Proof. By the aid of (2.3) we write 𝑓_𝑤𝑓•𝜙𝑥₂=𝑓•𝜙,𝐾_{𝜆,𝜇,𝑥2}𝑥₁. (4.2)
By using (3.1) we get 𝑓_𝑤𝑓•𝜙𝑥₂=𝑓𝜙⊙𝜏_𝑥2𝐾_{𝜆,𝜇,𝑥2}𝑥₁=𝜙𝑓⊙𝜏_𝑥2𝐾_{𝜆,𝜇,𝑥2}𝑥₁. (4.3)
That is 𝑓_𝑤(𝑓•𝜙)(𝑥₂)=𝜙⊙𝜏_𝑥2𝑇. Employing (4.1) yields 𝑓_𝑤(𝑓•𝜙)(𝑥₂)=(𝑇⋆𝜙)(𝑥₂), where 𝑇=𝑓_𝑤𝑓. This proves the theorem.

Lemma 4.2. Let 𝑇∈𝔽_𝑤(𝐑) and 𝜙∈𝔻(𝐑), and then 𝑇⋆𝜙∈𝔽_𝑤(𝐑).

Proof. Let 𝑓∈𝔼^′(𝐑) be such that 𝑇=𝑓_𝑤𝑓, and that then Theorem 4.1 implies 𝑇⋆𝜙=𝑓_𝑤𝑓•𝜙. (4.4)
Thus 𝑇⋆𝜙∈𝔽_𝑤(𝐑) by Lemma 3.3. Hence the lemma.

Lemma 4.3. Let 𝑇∈𝔽_𝑤(𝐑) and 𝜙∈𝔻(𝐑), and then 𝑓_𝑤⁻¹(𝑇⋆𝜙)=𝑓•𝜙.

Proof. Theorem 4.1 implies that 𝑓_𝑤⁻¹(𝑇⋆𝜙)=𝑓_𝑤⁻¹(𝑓_𝑤(𝑓•𝜙))=𝑓•𝜙. This completes the proof of the lemma.

Lemma 4.4. Let 𝑇₁,𝑇₂∈𝔽_𝑤(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) Let 𝑇_𝑛→𝑇 as 𝑛→∞ and 𝜙∈𝔻(𝐑), and then 𝑇_𝑛⋆𝜙→𝑇⋆𝜙 as 𝑛→∞ in 𝔽_𝑤(𝐑).
(ii) Let 𝑇_𝑛→𝑇 as 𝑛→∞ and (𝜙_𝑛)∈Δ, and then 𝑇_𝑛⋆𝜙_𝑛→𝑇 as 𝑛→∞ in 𝔽_𝑤(𝐑).

Proof. (i) Let 𝑇_𝑛,𝑇∈𝔽_𝑤(𝐑) then 𝑇=𝑓_𝑤𝑓  and 𝑇_𝑛=𝑓_𝑤𝑓_𝑛 for some 𝑓,𝑓_𝑛∈𝔼^′(𝐑). Hence, using Theorem 4.1, we have 𝑇_𝑛⋆𝜙=𝑓_𝑤𝑓_𝑛⋆𝜙=𝑓_𝑤𝑓_𝑛•𝜙. (4.5)
By Lemma 3.7 we get 𝑇_𝑛⋆𝜙→𝑓_𝑤𝑓•𝜙. (4.6)
Once again Theorem 4.1 implies that 𝑇_𝑛⋆𝜙→𝑓_𝑤𝑓⋆𝜙. (4.7) Thus we ahve 𝑇_𝑛⋆𝜙→𝑇⋆𝜙. (4.8)
This proves part (i) of the Lemma.
(ii) can be proved similarly by using Lemma 3.8 and Theorem 4.1. The space 𝔹_𝔽𝑤 is therefore established.

The sum of two Boehmians and multiplication by a scalar in 𝔹_𝔽𝑤 is defined in a natural way as follows: 𝑓_𝑛𝜙_𝑛+𝑔_𝑛𝜓_𝑛=𝑓_𝑛⋆𝜓_𝑛+𝑔_𝑛⋆𝜙_𝑛𝜙_𝑛⋆𝜓_𝑛,𝛼𝑓_𝑛𝜙_𝑛=𝛼𝑓_𝑛𝜙_𝑛,𝛼∈ℂ. (4.9)

The operation ⋆ and the differentiation are defined by 𝑓_𝑛𝜙_𝑛⋆𝑔_𝑛𝜓_𝑛=𝑓_𝑛⋆𝑔_𝑛𝜙_𝑛⋆𝜓_𝑛,𝐷^𝛼𝑓_𝑛𝜙_𝑛=𝐷^𝛼𝑓_𝑛𝜙_𝑛. (4.10)

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 [𝑓_𝑛/𝜙_𝑛]∈𝔹₁, and then 𝔉_𝑤⁻¹𝑇_𝑓𝑛𝜙_𝑛=𝑇_𝑓⁻¹_𝑛𝜙_𝑛,𝔉_𝑤𝑓_𝑛𝜙_𝑛=𝑇_𝑓𝑛𝜙_𝑛, (5.1) for each (𝜙_𝑛)∈Δ, where 𝑇_𝑓𝑛=𝑓_𝑤𝑓_𝑛.

Lemma 5.2. The optical Fresnel-wavelet transform 𝔉_𝑤∶𝔹₁→𝔹_𝔽𝑤 is well defined.

Proof. It is a straightforward.

Lemma 5.3. The optical Fresnel-wavelet transform 𝔉_𝑤∶𝔹₁→𝔹_𝔽𝑤 is linear.

Proof. It is straightforward by using Definition 5.1.

Lemma 5.4. The optical Fresnel-wavelet transform 𝔉_𝑤∶𝔹₁→𝔹_𝔽𝑤 is an isomorphism.

Proof. Assume that 𝔉_𝑤[𝑓_𝑛/𝜙_𝑛]=𝔉_𝑤[𝑔_𝑛/𝜓_𝑛], and then using (5.1) and the concept of quotients we get 𝑇_𝑓𝑛⋆𝜓_𝑚=𝑇_𝑔𝑚⋆𝜙_𝑛, where 𝑇_𝑓𝑛=𝑓_𝑤𝑓_𝑛 and 𝑇_𝑔𝑚=𝑓_𝑤𝑔_𝑚. Therefore, Theorem 4.1 implies that 𝑓_𝑤(𝑓_𝑛•𝜓_𝑚)=𝑓_𝑤(𝑔_𝑚•𝜙_𝑛). Properties of 𝑓_𝑤 imply that 𝑓_𝑛•𝜓_𝑚=𝑔_𝑚•𝜙_𝑛. Therefore, [𝑓_𝑛/𝜙_𝑛]=[𝑔_𝑛/𝜓_𝑛]. To establish 𝑓_𝑤 is surjective, and let [𝑇_𝑓𝑛/𝜙_𝑛]∈𝔹_𝐹𝑤. Then 𝑇_𝑓𝑛⋆𝜙_𝑚=𝑇_𝑓𝑚⋆𝜙_𝑛 for every 𝑚,𝑛∈ℕ. Hence 𝑓_𝑛,𝑓_𝑚∈𝔼^′(𝐑) are such that 𝑇_𝑓𝑛=𝑓_𝑤𝑓_𝑛 and 𝑇_𝑓𝑚=𝑓_𝑤𝑓_𝑚. Theorem 4.1 implies that 𝑓_𝑤(𝑓_𝑛•𝜙_𝑚)=𝑓_𝑤(𝑓_𝑚•𝜙_𝑛). Hence [𝑓_𝑛/𝜙_𝑛]∈𝔹₁ is such that 𝔉_𝑤[𝑓_𝑛/𝜙_𝑛]=[𝑇_𝑓𝑛/𝜙_𝑛]. This completes the proof of the lemma.

Now, Let [𝑓_𝑛/𝜙_𝑛]∈𝔹₁, and then we define the inverse optical Fresnel-wavelet transform of [𝑓_𝑛/𝜙_𝑛] as 𝔉_𝑤⁻¹𝑓_𝑛𝜙_𝑛=𝑇_𝑓⁻¹_𝑛𝜙_𝑛, (5.2) where 𝑇_𝑓⁻¹_𝑛=𝑓_𝑤⁻¹𝑓_n.

Lemma 5.5. Let [𝑇_𝑓𝑛/𝜙_𝑛]∈𝔹_𝔽𝑤, 𝑇_𝑓𝑛=𝑓_𝑤𝑓_𝑛, and 𝜙∈𝔻(ℝ) then 𝔉_𝑤⁻¹𝑇_𝑓𝑛𝜙_𝑛=𝑇⋆𝜙^𝑓−1𝑛𝜙_𝑛•𝜙,𝔉_𝑤𝑓_𝑛𝜙_𝑛=𝑇•𝜙^𝑓𝑛𝜙_𝑛⋆𝜙. (5.3)

Proof. Applying Definition 5.1 yields 𝔉_𝑤⁻¹𝑇_𝑓𝑛𝜙_𝑛⋆𝜙=𝔉_𝑤⁻¹𝑇_𝑓𝑛⋆𝜙𝜙_𝑛=𝑓_𝑤⁻¹𝑇_𝑓𝑛⋆𝜙𝜙_𝑛. (5.4)
Using Lemma 4.3 we obtain 𝔉_𝑤⁻¹𝑇_𝑓𝑛𝜙_𝑛=𝑓⋆𝜙^𝑤−1𝑇_𝑓𝑛𝜙_𝑛𝑇•𝜙=^𝑓−1𝑛𝜙_𝑛•𝜙. (5.5)

This completes the proof of the Lemma.

Theorem 5.6. 𝔉_𝑤∶𝔹₁→𝔹_𝔽𝑤 and 𝔉_𝑤⁻¹∶𝔹_𝔽𝑤→𝔹₁ are continuous with respect to 𝛿 and Δ convergences.

Proof. First of all, we show that 𝔉_𝑤∶𝔹₁→𝔹_𝔽𝑤 and 𝔉_𝑤⁻¹∶𝔹_𝔽𝑤→𝔹₁ are continuous with respect to 𝛿 convergence.
Let 𝛽→−_𝑛^𝛿𝛽 in 𝔹₁ as 𝑛→∞, and then we show that 𝔉_𝑤𝛽_𝑛→𝔉_𝑤𝛽 as 𝑛→∞. By virtue of [25] we can find 𝑓_𝑛,𝑘 and 𝑓_𝑘 in 𝔼^′(𝐑) such that 𝛽_𝑛=𝑓_𝑛,𝑘𝜙_𝑘𝑓,𝛽=^𝑘𝜙_𝑘 (5.6)
such that 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ for every 𝑘∈ℕ. Employing the continuity condition of the optical Fresnel-wavelet transform implies that 𝑇_{𝑓𝑛,𝑘}→𝑇_𝑓𝑘 as 𝑛→∞ in the space 𝔽_𝑤(𝐑). Thus, [𝑇_{𝑓𝑛,𝑘}/𝜙_𝑘]→[𝑇_𝑓𝑘/𝜙_𝑘] as 𝑛→∞ in 𝔹_𝔽𝑤.
To prove the second part, let 𝑔→−_𝑛^𝛿𝑔 in 𝔹_𝔽𝑤 as 𝑛→∞. Then, once again, by [4], 𝑔_𝑛=[𝑇_{𝑓𝑛,𝑘}/𝜙_𝑘] and 𝑔=[𝑇_𝑓𝑘/𝜙_𝑘] for some 𝑇_{𝑓𝑛,𝑘},𝑇_𝑓𝑘∈𝔽_𝑤(𝐑) and 𝑇_{𝑓𝑛,𝑘}→𝑇_𝑓𝑘 as 𝑛→∞. Hence 𝑓_𝑤⁻¹𝑇_{𝑓𝑛,𝑘}→𝑓_𝑤⁻¹𝑇_𝑓𝑘 in 𝔹₁ as 𝑛→∞. Or, [𝑇_𝑓⁻¹_𝑛,𝑘/𝜙_𝑘]→[𝑇_𝑓⁻¹_𝑘/𝜙_𝑘] as 𝑛→∞. Using Definition 5.1 we get 𝔉_𝑤⁻¹[𝑇_{𝑓𝑛,𝑘}/𝜙_𝑘]→𝔉_𝑤⁻¹[𝑇_𝑓𝑘/𝜙_𝑘] as 𝑛→∞.
Now, we establish continuity of 𝔉_𝑤 and 𝔉_𝑤⁻¹ with respect to Δ convergence. Let 𝛽−→_𝑛^Δ𝛽 in 𝔹₁ as 𝑛→∞. Then, there exist 𝑓_𝑛∈𝔼^′(𝐑) and 𝜙_𝑛∈Δ such that (𝛽_𝑛−𝛽)•𝜙_𝑛=[(𝑓_𝑛•𝜙_𝑘)/𝜙_𝑘] and 𝑓_𝑛→0 as 𝑛→∞. Employing Definition 5.1 we get 𝔉_𝑤𝛽_𝑛−𝛽•𝜙_𝑛=𝑓_𝑤𝑓_𝑛•𝜙_𝑘𝜙_𝑘. (5.7)
Hence, from Lemma 4.2 we have 𝔉_𝑤((𝛽_𝑛−𝛽)•𝜙_𝑛)=[(𝑇_𝑓𝑛⋆𝜙_𝑘)/𝜙_𝑘]=𝑇_𝑓𝑛→0 as 𝑛→∞ in 𝔽_𝑤. Therefore consider 𝔉_𝑤𝛽_𝑛−𝛽•𝜙_𝑛=𝔉_𝑤𝛽_𝑛−𝔉_𝑤𝛽⋆𝜙_𝑛→0as𝑛→∞. (5.8)
Hence,𝔉_𝑤𝛽−→_𝑛^Δ𝔉_𝑤𝛽 as 𝑛→∞. Finally, let 𝑔−→_𝑛^Δ𝑔 in 𝔹_𝔽𝑤 as 𝑛→∞, and then we find 𝑇_𝑓𝑘∈𝔽_𝑤(𝐑) such that (𝑔_𝑛−𝑔)⋆𝜙_𝑛=[(𝑇_𝑓𝑘⋆𝜙_𝑘)/𝜙_𝑘] and 𝑇_𝑓𝑘→0 as 𝑛→∞ for some (𝜙_𝑛)∈Δ and 𝑇_𝑓𝑘=𝑓_𝑤𝑓_𝑛. Now, using Definition 5.1, we obtain that 𝔉_𝑤⁻¹𝑔_𝑛−𝑔⋆𝜙_𝑛=𝑓_𝑤⁻¹𝑇_𝑓𝑘⋆𝜙_𝑘𝜙_𝑘. (5.9)
Lemma 5.5 implies that 𝔉_𝑤⁻¹𝑔_𝑛−𝑔⋆𝜙_𝑛=𝑓_𝑛•𝜙_𝑘𝜙_𝑘=𝑓_𝑛→0as𝑛→∞in𝔼^′(𝐑). (5.10)
Thus we have 𝔉_𝑤⁻¹𝑔_𝑛−𝑔⋆𝜙_𝑛=𝔉_𝑤⁻¹𝑔_𝑛−𝔉_𝑤⁻¹𝑔•𝜙_𝑛→0as𝑛→∞. (5.11)
From this we find that 𝔉⁻¹_w𝑔−→_𝑛^Δ𝔉_𝑤⁻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.