1 Introduction

The Fresnel integral operator is a paraxial diffraction integral operator, which describes the propagation of light from one transverse plane along the optical axis to another. In the one-dimensional case, it has been defined by aid of the integral equation (see, e.g., [1])

$$ \widehat{v} ( x ) =e^{i2\pi d/\lambda }e^{-i\pi /4}\sqrt{ \frac{1}{\lambda d}} \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi ( x-\acute{x} ) ^{2}}{\lambda d} \biggr) v ( \acute{x} ) \,d \acute{x}, $$
(1)

where v̂ is the output field related to the input field v, d is the distance of the propagation of light, and λ is the length of the wave. When \(v_{a}\) specifies the fractional Fourier integral operator with order parameter a, the Fresnel integral operator can be explicitly decomposed into the fractional Fourier integral operator \(v_{a}\) followed by magnification and chirp multiplication as follows (see, e.g., [2]):

$$ \widehat{v} ( x ) =e^{i2\pi d/\lambda }e^{-ia\pi /4}\sqrt{ \frac{1}{sM}}\exp \biggl( \frac{i\pi x^{2}}{\lambda \acute{R}} \biggr) v_{a } \biggl( \frac{x}{sM} \biggr) , $$

where

$$ a=\frac{2}{\pi }\arctan \frac{\lambda d}{s^{2}},\qquad M=\sqrt{1+ \frac{\lambda ^{2}d^{2}}{s^{4}}}=\sec \frac{a\pi }{2}\quad \text{and}\quad \acute{R}=\frac{s^{4}+\lambda ^{2}d^{2}}{\lambda ^{2}d}=d\csc ^{2}\frac{a\pi }{2}. $$

In the last era, the Fresnel integral operator has been implemented widely in Fourier optics (see, e.g., [2]), phase retrieval techniques (see, e.g., [3–5]), digital holography (see, e.g., [6–9] and [10–13]), biological imaging applications (see, e.g., [14, 15]), beam shaping (see, e.g., [16, 17]), deformations (see, e.g., [18–20]), and many other areas of integral inequalities and the porous medium (see, e.g., [21–23]).

The name Boehmians is used to describe a space of objects that are defined as equivalence classes of pairs of sequences. Boehmians are constructed by using a set of axioms with two notions of convergence called δ-convergence and Δ-convergence. One of these axioms states that the denominator sequences are set to form a commutative semigroup with respect to a binary operation. To a context of Boehmians, certain variants of the Fresnel operator were extended to classes of Boehmians. In [24], a generalized Fresnel integral operator was discussed on a class of distributions and a space of integrable Boehmians. In [25], a diffraction Fresnel integral operator was discussed on strong Boehmian spaces. Likewise, in literature, various integral operators were discussed and thoroughly applied to various spaces of Boehmians, see for example [6, 10, 11, 26–37] and the references cited therein. However, in this approach, we discuss a new version of paraxial diffraction operators. We derive convolution products, convolution theorems, sets of Boehmians, and certain generalization of the diffraction operator. In contrast to profitable approaches furnished in [24] and [25], our results are new and distinguishable. In Sect. 2, we introduce convolution products and delta sequences, and extract various results. In Sect. 3, we provide several axioms and generate appropriate sets of Boehmians. In Sect. 4, we derive the generalized Fresnel integral and examine its desired properties.

2 Convolution theorem and delta sequences

In this section, we make use of the integral formula of the Fresnel operator and derive meritorious convolution products and delta sequences. On this basis, we consequently implement our results in constructing the integral convolution theorem. The very prerequisite convolution product to this variant can be earned as follows.

Definition 1

Let \(v_{1}\) and \(v_{2}\) be integrable functions in the space \(L^{1}\). Then the product formula ∗ between \(v_{1}\) and \(v_{2}\) is defined as follows:

$$ v_{1}\ast v_{2} ( \acute{x} ) = \int _{-\infty }^{ \infty }v_{1} ( w ) v_{2} ( \acute{x}-w ) W ( \acute{x},w ) \,dw, $$
(2)

where

$$ W ( \acute{x},w ) =\exp \biggl( 2iw \frac{\pi ( w-\acute{x} ) }{\lambda d} \biggr) $$

is the proposed weight function, d is the distance of the propagation of light, and λ is the length of the wave.

Denote by \(L^{1}\) the set of all integrable functions on \(\mathbb{R}\). Then, for the convenience of the reader, we drop the leading constant phase term \(L=\exp ( i2\pi d/\lambda ) \exp ( -i\pi /4 ) \) for the rest of the analysis.

The convolution theorem of the classical Fresnel integral can be drawn as follows.

Theorem 2

Let\(v_{1}\)and\(v_{2}\)be two integrable functions in\(L^{1}\). Then the Fresnel integral operator of the convolution product\(v_{1}\ast v_{2}\)is given by

$$ \widehat{v_{1}\ast v_{2}} ( x ) = \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \widehat{v}_{1} ( x ) \widehat{v}_{2} ( x ) , $$

where\(L=\exp ( i2\pi d/\lambda ) \exp ( -i\pi /4 ) \).

Proof

Let \(v_{1}\) and \(v_{2}\) be integrable functions over \(\mathbb{R} \). Then, by applying (1), we routinely write

$$ \widehat{v_{1}\ast v_{2}} ( x ) = \int _{-\infty }^{ \infty }\exp \biggl( \frac{i\pi ( x-\acute{x} ) ^{2}}{\lambda d} \biggr) ( v_{1}\ast v_{2} ) ( \acute{x} ) \,d\acute{x}. $$

Hence, in view of (2), we get

$$ \widehat{v_{1}\ast v_{2}} ( x ) = \int _{-\infty }^{ \infty }\exp \biggl( \frac{i\pi ( x-\acute{x} ) ^{2}}{\lambda d} \biggr) \int _{-\infty }^{ \infty }v_{1} ( w ) v_{2} ( \acute{x}-w ) W ( \acute{x},w ) \,dw\,d\acute{x}. $$
(3)

By taking into account the definition of W and using the change of variables \(z=\acute{x}-w\) together with Fubini’s theorem, we modify (3) as

$$ \widehat{v_{1}\ast v_{2}} ( x ) = \int _{-\infty }^{ \infty }\widehat{v}_{1} ( w ) \int _{-\infty }^{\infty } \exp \biggl( \frac{i\pi ( z^{2}+w^{2}-2 ( z+w ) x+x^{2} ) }{\lambda d} \biggr) \widehat{v}_{2} ( z ) \,dw\,dz. $$

Therefore, modifying and splitting out the exponent term under the integral sign in the previous equation yield

$$\begin{aligned} \widehat{v_{1}\ast v_{2}} ( x ) =& \int _{- \infty }^{\infty }v_{1} ( w ) \int _{-\infty }^{ \infty }\exp \biggl( \frac{i\pi ( ( z+w ) ^{2}+ ( w-x ) ^{2}-x^{2} ) }{\lambda d} \biggr) v_{2} ( z ) \,dz\,dw \\ =&e^{-i\pi x^{2}/\lambda d} \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi ( w-x ) ^{2}}{\lambda d} \biggr) v_{1} ( w ) \,dw \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi ( z-x ) ^{2}}{\lambda d} \biggr) v_{2} ( z ) \,dz. \end{aligned}$$

Hence, by taking into account the value of the leading constant phase term, we, indeed, obtain

$$ \widehat{v_{1}\ast v_{2}} ( x ) = \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \widehat{v}_{1} ( x ) \widehat{v}_{2} ( x ) . $$

Hence, the proof of the theorem has been completed. □

For deterministic needs, we establish the following very useful theorem.

Theorem 3

Let\(v_{1}\)and\(v_{2}\)be two integrable functions in\(L^{1}\). Then the convolution product\(v_{1}\ast v_{2}\)is an integrable function in\(L^{1}\).

Proof

Let \(v_{1}\) and \(v_{2}\) be in \(L^{1}\). Then, by aid of the fact that \(\vert W ( \acute{x},w ) \vert \leq 1\), we write

$$ \int _{-\infty }^{\infty } \bigl\vert v_{1}\ast v_{2} ( \acute{x} ) \bigr\vert \,d\acute{x}\leq \int _{-\infty }^{ \infty } \int _{-\infty }^{\infty } \bigl\vert v_{1} ( w ) \bigr\vert \bigl\vert v_{2} ( \acute{x}-w ) \bigr\vert \,dw\,d \acute{x}. $$

Hence, by using Fubini’s theorem, the previous integral inequality yields

$$ \int _{-\infty }^{\infty } \bigl\vert v_{1}\ast v_{2} ( \acute{x} ) \bigr\vert \,d\acute{x}\leq \int _{-\infty }^{ \infty } \bigl\vert v_{1} ( w ) \bigr\vert \int _{-\infty }^{\infty } \bigl\vert v_{2} ( \acute{x}-w ) \bigr\vert \,d\acute{x}\,dw< \infty . $$

The proof of the theorem is, therefore, completed. □

The fact which deserves further attention is that \(\widehat{v}\in L^{1}\) for every \(v\in L^{1}\). Based upon this conjecture, we may think about the delta sequence choice. Indeed, by following (1), we deduce that

$$ \widehat{v} ( 0 ) =L \int _{-\infty }^{\infty } \exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) v ( \acute{x} )\, d\acute{x}. $$

Therefore, a wise choice of the description of delta sequences can be expressed in the following way.

Definition 4

Let D denote the Schwartz space of test functions of compact supports over \(\mathbb{R}\). By Δ we denote the collection of all sequences from D such that

$$\begin{aligned}& \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) w_{n} ( \acute{x} ) \,d\acute{x}=1. \end{aligned}$$
(4)
$$\begin{aligned}& \int _{-\infty }^{\infty } \biggl\vert \exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) w_{n} ( \acute{x} ) \biggr\vert \,d \acute{x}< A,\quad A< \infty . \end{aligned}$$
(5)
$$\begin{aligned}& \operatorname{supp} ( w_{n} ) (\acute{x} ) \subseteq ( -a_{n},a_{n} ) , \quad a_{n} \rightarrow 0\text{ as }n\rightarrow \infty , \end{aligned}$$
(6)

where \(\operatorname{supp} ( w )\) is the support function of w. The following assertion holds for the delta sequences.

Theorem 5

Let\(( w_{n} )\)and\(( u_{n} )\)be two sequences in Δ. Then\(( w_{n}\ast u_{n} )\)is a sequence in Δ.

Proof

Firstly, we show that the integral equation

$$ \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) (w_{n}\ast u_{n} ) ( \acute{x} ) \,d\acute{x}=1 $$

is satisfied for Δ sequences. By Theorem 2 we are allowed to write

$$ \widehat{w_{n}\ast u_{n}} ( 0 ) =\frac{1}{L} \widehat{w}_{n} ( 0 ) \widehat{u}_{n} ( 0 ) . $$
(7)

Therefore, using (4), (7) may give

$$\begin{aligned}& \begin{aligned}[b] L \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) ( w_{n}\ast u_{n} ) ( \acute{x} )\, d\acute{x} ={}&\frac{1}{L} \biggl( L \int _{-\infty }^{\infty } \exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) w_{n} ( \acute{x} ) \,d\acute{x} \biggr) \\ &{}\times\biggl( L \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) u_{n} ( \acute{x} ) \,d\acute{x} \biggr) . \end{aligned} \end{aligned}$$
(8)

Hence, owing to the fact that \((w_{n} ) \) and \(( u_{n} ) \) are delta sequences in Δ, (8) then reveals

$$ \int _{-\infty }^{\infty }\exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) ( w_{n}\ast u_{n} ) ( \acute{x} ) d\acute{x}=1 \quad \text{for all } n\in \mathbb{N}. $$

This proves (4). Proofs of (5) and (6), namely,

$$ \int _{-\infty }^{\infty } \biggl\vert \exp \biggl( \frac{i\pi \acute{x}^{2}}{\lambda d} \biggr) ( w_{n}\ast u_{n} ) ( \acute{x} ) \biggr\vert d\acute{x}< A,\quad A< \infty , $$

and

$$ \operatorname{supp} ( w_{n}\ast u_{n} ) ( \acute{x} ) \subseteq ( -a_{n},a_{n} ) ,\quad a_{n} \rightarrow 0 \text{ as } n\rightarrow \infty , $$

respectively, are straightforward results from the definitions. Hence, we delete the details of the similar proofs. □

This completes the proof of the theorem.

3 Fresnel integrable spaces of Boehmians

We devote this section to the abstract construction of the space \(B^{S}\) of integrable Boehmians and the space \(H^{S}\) of ultra-Boehmians. We make a free use of the convolution theorem and define the convolution product which works in with the convolution ∗. To derive the generalized space \(B^{S}\) of integrable Boehmians, we establish the following exemplary axioms.

Theorem 6

The following identities hold:

  1. (i)

    \(( v_{1}+v_{2} ) \ast u_{n}=v_{1}\ast u_{n}+v_{2} \ast u_{n}\), where\(v_{1},v_{2}\in L^{1}\), \(n\in \mathbb{N}\), and\(u_{n}\in D\).

  2. (ii)

    \(( \alpha v ) \ast u_{n}=\alpha ( v\ast u_{n} ) \), where\(u_{n}\in D\), \(v\in L^{1}\), \(n\in \mathbb{N}\), and\(\alpha \in \mathbb{C}\).

  3. (iii)

    If\(v_{n}\rightarrow v\)as\(n\rightarrow \infty \)in\(L^{1}\)and\(u_{n}\in D\), then\(v_{n}\ast \psi \rightarrow v\ast \psi \)as\(n\rightarrow \infty \).

  4. (iv)

    \(v_{1}\ast v_{2}=v_{2}\ast v_{1}\)for\(v_{1},v_{2}\in L^{1} \).

  5. (v)

    \(v_{1}\ast ( u_{1}\ast u_{2} ) = ( v_{1} \ast u_{1} ) \ast u_{2}\), where\(v_{1}\in L^{1}\)and\(u_{1},u_{2}\in D\).

Proof

As the proofs of parts (i), (ii), and (iii) are straightforward consequences, following from simple integration, we prove part (iv) and part (v). By the convolution theorem, we get

$$ \widehat{v_{1}\ast v_{2}}= \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{v}_{1}\widehat{v}_{2}= \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \widehat{v}_{2} \widehat{v}_{1}=\widehat{v_{2} \ast v_{1}}. $$

Hence, by considering the inverse integral operator for both sides of the preceding equation, we complete the proof of (iv). The proof of part (v) is similar to the proof of part (iv). Therefore, we delete the details.

Hence, the proof of the theorem is completed. □

Theorem 7

If\(v\in L^{1}\)and\(( w_{n} ) \in \Delta \), then\(v\ast w_{n}\rightarrow v\)as\(n\rightarrow \infty \)in\(L^{1}\).

Proof of this theorem follows from similar analysis given by the same author (see, e.g., [24, 25]).

The Boehmian space \(B^{S}\) is based on the sets \(L^{1}\), D, and Δ, and the convolution product ∗ is obtained. A Boehmian in \(B^{S}\) may be introduced as

$$ x_{v_{n}}= \biggl( \frac{v_{n}}{w_{n}} \biggr) , $$

where \(v_{n}\in L^{1}\) and \(( w_{n} ) \) is a delta sequence. If \(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \) and \(x_{u_{n}}= ( \frac{u_{n}}{\varepsilon _{n}} ) \) are in \(B^{S}\), then it is possible to claim the space operations

$$ x_{v_{n}}+x_{u_{n}}= \biggl( \frac{v_{n}\ast w_{n}+u_{n}\ast w_{n}}{w_{n}\ast \varepsilon _{n}} \biggr) \quad \text{and}\quad \beta x_{v_{n}}= \biggl( \frac{\beta v_{n}}{w_{n}} \biggr) ,\quad \forall \beta \in \mathbb{C}. $$

For every \(k\in \mathbb{N}\) and \(\psi \in L^{1}\), we define on \(B^{S}\) the following useful operations:

$$ x_{v_{n}}\ast x_{u_{n}}= \biggl( \frac{v_{n}\ast u_{n}}{w_{n}\ast \varepsilon _{n}} \biggr),\qquad D^{k}x_{v_{n}}= \biggl( \frac{D^{k}v_{n}}{w_{n}} \biggr)\quad \text{and}\quad x_{v_{n}}\ast \psi = \biggl( \frac{v_{n}\ast \psi }{w_{n}} \biggr) , $$

where \(D^{k}x_{v_{n}}\) is the kth derivative of \(x_{v_{n}}\).

Definition 8

Let \(x_{v_{n},m},x_{v_{n}}\in B^{S}\), \(m=1,2,3,\ldots \) . Then the sequence \(( x_{v_{n},m} ) _{m=1}^{\infty }\) is δ-convergent to \(x_{v_{n}}\), denoted by \(\delta -\lim_{m\rightarrow \infty }x_{v_{n},m}=x_{v_{n}}\), provided that there can be found a delta sequence \(( w_{n} ) \) such that

  1. (i)

    \(( x_{v_{n},m}\ast w_{k} ) , ( x_{v_{n}} \ast w_{k} ) \in L^{1}\) for all \(m,k\in \mathbb{N}\), and

  2. (ii)

    \(\lim_{m\rightarrow \infty }x_{v_{n},m}\ast _{\alpha ,\beta }w_{k}=x_{v_{n}} \ast w_{k}\) in \(L^{1}\) for every \(k\in \mathbb{N}\).

Or, equivalently,

\(\delta -\lim_{m\rightarrow \infty }x_{v_{n},m}=x_{v_{n}}\) if and only if there are \(u_{n,k}\), \(u_{k}\in L^{1}\) and \(( w_{k} ) \in \Delta \) such that

  1. (i)

    \(w_{n}= ( \frac{u_{n,k}}{w_{k}} ) \), \(x_{v_{n}}= ( \frac{u_{k}}{w_{k}} ) \)

  2. (ii)

    to every \(k\in N\), we have \(\lim_{n\rightarrow \infty }u_{n,k}=u_{k}\) in \(L^{1}\).

Definition 9

Let \(x_{v_{n},m},x_{v_{n}}\in B^{S} \), \(m=1,2,3,\ldots \) . Then the sequence \(( x_{v_{n},m} ) _{m=1}^{\infty }\) is Δ-convergent to \(x_{v_{n}}\), denoted by Δ-\(\lim_{m\rightarrow \infty }x_{v_{n},m}=x_{v_{n}}\), provided that there can be found a delta sequence \(( w_{n} ) \) such that

  1. (a)

    \(( x_{v_{n},m}-x_{v_{n}} ) \ast w_{n}\in L^{1}\)\(( \forall m\in \mathbb{N} ) \);

  2. (b)

    \(\lim_{m\rightarrow \infty } ( x_{v_{n},m}-x_{v_{n}} ) \ast w_{n}=0\) in \(L^{1}\).

Theorem 10

Let\(( w_{n} ) \in \Delta \)be fixed and\(u\in L^{1}\). Then the mapping

$$ u\rightarrow x_{v_{n}}, $$
(9)

\(x_{v_{n}}= ( \frac{u\ast w_{n}}{w_{n}} ) \), is a one-one map from the space\(L^{1}\)into the Boehmian space\(B^{S}\).

From (9), it can be read that \(L^{1}\) has an identification with a subspace in \(B^{S}\).

Theorem 11

Let\(( w_{n} ) \in \Delta \). If\(u_{n}\rightarrow u\)in\(L^{1}\)as\(n\rightarrow \infty \), then, for all\(k\in N\), \(u_{n}\ast u_{k}\rightarrow u\ast u_{k}\)as\(n\rightarrow \infty \).

Theorem 11 clearly shows that

$$ x_{v_{n},m}\rightarrow x_{v_{n}}\quad \text{in } B^{S} \text{ as } m \rightarrow \infty . $$

Moreover, the above theorem indeed leads to the following statement.

Theorem 12

The embedding of the space\(L^{1}\)into the Boehmian space\(B^{S}\)defined by (9) is continuous.

Now, let us evaluate the ultra-Boehmian space \(H^{S}\). Introduce the set \(I^{S}\) to be the set of all Fresnel operators of all \(L^{1}\) elements. Similarly, denote by \(D^{S} ( \text{ or }\Delta ^{S} ) \) the set of all Fresnel transforms of all \(D ( \text{ or }\Delta ) \) elements, respectively. The product formula ▷ is defined for \(H^{S}\) as follows.

Definition 13

Let \(\widehat{v}\in I^{S}\) and \(u^{S}\in D^{S}\). Then we define

$$ \widehat{v}\triangleright \widehat{u}= \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{v} \widehat{u}. $$
(10)

We prove the following necessary axioms.

Theorem 14

The following hold:

  1. (i)

    \(( \widehat{v}_{1}+\widehat{v}_{2} ) \triangleright \widehat{u}=\widehat{v}_{1}\triangleright \widehat{u}+\widehat{v}_{2}\triangleright \widehat{u}\)for every\(\widehat{v}_{1},\widehat{v}_{2}\in I^{S}\)and\(\widehat{u}\in D^{S}\).

  2. (ii)

    \(( \beta \widehat{v} ) \triangleright \widehat{u}=\beta ( \widehat{v}\triangleright \widehat{u} ) \)for every\(\widehat{v}_{2}\in I^{S}\), \(\widehat{u}\in D^{S}\), and\(\beta \in \mathbb{C}\); the space of complex numbers.

  3. (iii)

    If\(\widehat{v}_{n}\rightarrow \widehat{v}\)as\(n\rightarrow \infty \)and\(\widehat{u}\in D^{S}\), then\(\widehat{v}_{n}\triangleright \widehat{u}\rightarrow \widehat{v}\triangleright \widehat{u}\)as\(n\rightarrow \infty \)in\(I^{S}\).

  4. (iv)

    \(\widehat{v}_{1}\triangleright \widehat{v}_{2}= \widehat{v}_{2}\triangleright \widehat{v}_{1}\)for\(\widehat{v}_{1},\widehat{v}_{2}\in I^{S}\).

  5. (v)

    If\(\widehat{v}\in I^{S} \)and\(( \widehat{w}_{n} ) \in \Delta ^{S}\), then\(\widehat{v}\triangleright \widehat{w}_{n}\rightarrow \widehat{v}\)in\(I^{S}\)as\(n\rightarrow \infty \).

Proof

Proof of (i). Let \(\widehat{v}_{2}\in I^{S}\) and \(\widehat{u}\in D^{S}\) be the Fresnel operator of the elements \(u\in L^{1}\) and \(v\in D\), respectively. Then, by (10), we have

$$\begin{aligned} ( \widehat{v}_{1}+\widehat{v}_{2} ) \triangleright \widehat{u} =&\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} ( \widehat{v}_{1}+\widehat{v}_{2} ) \widehat{u} \\ =&\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{v}_{1}\widehat{u}+ \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \widehat{v}_{2}\widehat{u} \\ =&\widehat{v}_{1}\triangleright \widehat{u}+\widehat{v}_{2} \triangleright \widehat{u}. \end{aligned}$$

Proof of (ii) follows from a similar argument used in the proof of (i). To prove (iii), let \(\widehat{u}\in D^{S}\) be the Fresnel operator of \(u\in D\) and \(\widehat{v}_{n}\), v̂ be the Fresnel operators of \(v_{n}\) and \(v\in L^{1}\), respectively. Then, by (10) and Theorem 2, we proceed to write

$$\begin{aligned} ( \widehat{v}_{n}\triangleright \widehat{u}-\widehat{v} \triangleright \widehat{u} ) =&\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{v}_{n}\widehat{u}- \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{v} \widehat{u} \\ =&\widehat{v_{n}\ast u}-\widehat{v\ast u} \\ =&\widehat{v_{n}\ast u-v\ast u} \\ =&\widehat{ ( v_{n}-v ) \ast u} \\ =&\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \widehat{v_{n}-v}\widehat{u}. \end{aligned}$$

Hence, by the assumption that \(\widehat{v}_{n}\rightarrow \widehat{v}\) as \(n\rightarrow \infty \), the right-hand side of the above equation approaches zero as \(n\rightarrow \infty \). Hence, we have obtained \(\widehat{v}_{n}\triangleright \widehat{u}\rightarrow \widehat{v}\triangleright \widehat{u}\) as \(n\rightarrow \infty \). The proofs of (iv) and (v) are similar.

This completes the proof of the theorem. □

The Boehmian space \(H^{S}\) is approved by the above theorem. Every member of \(H^{S}\) is defined as

$$ x_{\widehat{v}_{n}}= \biggl( \frac{\widehat{v}_{n}}{\widehat{w}_{n}} \biggr) . $$

If \(x_{\widehat{v}_{n}}= ( \frac{\widehat{v}_{n}}{\widehat{w}_{n}} ) \) and \(x_{\widehat{u}_{n}}= ( \frac{\widehat{u}_{n}}{\widehat{\varepsilon }_{n}} ) \) are in \(H^{S}\), then we define

$$ x_{\widehat{v}_{n}}+x_{\widehat{u}_{n}}= \frac{\widehat{v}\triangleright \widehat{w}_{n}+\widehat{u}_{n}\triangleright \widehat{w}_{n}}{\widehat{w}_{n}\triangleright \widehat{\varepsilon }_{n}}\quad \text{and}\quad \beta x_{ \widehat{v}_{n}}= \biggl( \frac{\beta \widehat{v}}{\widehat{w}_{n}} \biggr) ,\quad \forall \beta \in \mathbb{C}. $$

Also, for \(k\in \mathbb{R}\) and \(\hat{\psi }\in I^{S}\), we define operations on \(H^{S}\) as

$$ x_{\widehat{v}_{n}}\triangleright x_{\widehat{u}_{n}}= \biggl( \frac{\widehat{v}\triangleright \widehat{u}_{n}}{\widehat{w}_{n}\triangleright \widehat{\varepsilon }_{n}} \biggr) ,\qquad D^{k}x_{\widehat{v}_{n}}= \biggl( \frac{D^{k}\widehat{v}}{\widehat{w}_{n}} \biggr) \quad \text{and}\quad x_{\widehat{v}_{n}}\triangleright \hat{\psi }= \biggl( \frac{\widehat{v}\triangleright \hat{\psi }}{\widehat{w}_{n}} \biggr) , $$

where \(D^{k}x_{\widehat{v}_{n}}\) is the kth derivative of \(x_{\widehat{v}_{n}}\).

Definition 15

Let \(x_{\widehat{v}_{n},m},x_{\widehat{v}_{n}}\in H^{S}\) for \(m=1,2,3,\ldots \) . Then the convergence in \(H^{S} \) is defined as follows.

  1. (i)

    The sequence \(( x_{\widehat{v}_{n},m} ) _{m=1}^{\infty }\) is said to be δ-convergent to \(x_{\widehat{v}_{n}}\), denoted by \(\delta -\lim_{m\rightarrow \infty }x_{\widehat{v}_{n},m}=x_{ \widehat{v}_{n}}\), provided that there can be found a delta sequence \(( \widehat{w}_{n} ) \) such that

    1. (a)

      \(( x_{\widehat{v}_{n},m}\triangleright w_{k} ) , ( x_{\widehat{v}_{n}}\triangleright \widehat{w}_{k} ) \in I^{S}\) for all \(n,k\in \mathbb{N}\).

    2. (b)

      \(\lim_{m\rightarrow \infty }x_{\widehat{v}_{n},m}\triangleright \widehat{w}_{k}=x_{\widehat{v}_{n}} \triangleright \widehat{w}_{k}\) in \(I^{S}\) for every \(k\in \mathbb{N}\).

  2. (ii)

    The sequence \(( x_{\widehat{v}_{n},m} ) _{m=1}^{\infty }\) is \(\Delta ^{S}\)-convergent to \(x_{\widehat{v}_{n}}\), denoted by \(\Delta ^{S}\)-\(\lim_{m\rightarrow \infty }x_{\widehat{v}_{n},m}=x_{ \widehat{v}_{n}}\), provided that there can be found a delta sequence \(( \widehat{w} ) \) such that

    1. (a)

      \(( x_{\widehat{v}_{n},m}-x_{\widehat{v}_{n}} ) \triangleright \widehat{w}_{n}\in I^{S}\)\(( \forall m\in \mathbb{N} ) \);

    2. (b)

      \(\lim_{m\rightarrow \infty } ( x_{\widehat{v}_{n},m}-x_{\widehat{v}_{n}} ) \triangleright \widehat{w}_{n}=0\) in \(I^{S}\).

Theorem 16

Let\(( \widehat{w} ) \in \Delta ^{S}\)be fixed and\(u\in I^{S}\). Then the mapping defined by

$$ u\rightarrow x_{\widehat{v}_{n}}, $$
(11)

where\(x_{\widehat{v}_{n}}= ( \frac{u\triangleright \widehat{w}}{\widehat{w}} ) \), is a one-to-one mapping\(I^{S}\rightarrow H^{S}\).

From (11), it can be read that \(I^{S}\) can be identified as subspaces of \(H^{S}\).

Theorem 17

Let\(( \widehat{w} ) \in \Delta ^{S} \). Then if\(u_{n}\rightarrow u\)in\(I^{S}\)as\(n\rightarrow \infty \), then, for all\(k\in \mathbb{N}\), \(u_{n}\triangleright u_{k}\rightarrow u \triangleright u_{k}\)as\(n\rightarrow \infty \). That is, \(x_{\widehat{v}_{n},m}\rightarrow x_{\widehat{v}_{n}}\)in\(H^{S}\)as\(m\rightarrow \infty \).

The above theorem leads to the following:

Theorem 18

The mapping defined by (9) is a continuous embedding of the space\(I^{S}\)into the ultra-Boehmian space\(H^{S}\).

4 The boehmian Fresnel operators

By the former analysis, we define the Fresnel operator of the Boehmian \(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \) in \(B^{S}\) as a Boehmian in \(H^{S}\) defined by

$$ \widehat{x_{v_{n}}}=x_{\widehat{v}_{n}}, $$
(12)

where \(x_{\widehat{v}_{n}}= ( \frac{\widehat{v}_{n}}{\widehat{w}_{n}} ) \).

Theorem 19

Let\(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \). Then the mapping\(x_{v_{n}}\rightarrow x_{\widehat{v}_{n}}\)from\(B^{S}\rightarrow H^{S}\)defined by\(\widehat{x_{v_{n}}}=x_{\widehat{v}_{n}} \)coincides with the classical Fresnel operator\(L^{1}\rightarrow I^{S}\).

Proof

Let \(w\in L^{1}\). Then w can be identified in \(B^{S}\) as \(x_{v_{n}}\), where \(x_{v_{n}}= ( \frac{w\ast v_{n}}{v_{n}} ) \), which is the representation of w in \(B^{S}\). Now, by employing Theorem 2, we obtain

$$ x_{\widehat{v}_{n}}= \biggl( \frac{\widehat{w\ast v_{n}}}{\widehat{v}_{n}} \biggr) = \frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} \biggl( \frac{\hat{w}\widehat{v}_{n}}{\widehat{v}_{n}} \biggr) = \biggl( \frac{\hat{w}\triangleright \widehat{v}_{n}}{\widehat{v}_{n}} \biggr), $$

which is the representation of ŵ in \(H^{S}\).

The proof is, therefore, finished. □

Theorem 20

Let\(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \)and\(\widehat{x_{v_{n}}}=x_{\widehat{v}_{n}}\). Then the mapping\(x_{v_{n}}\rightarrow x_{\widehat{v}_{n}}\)from\(B^{S}\)onto\(H^{S}\), defined by\(\widehat{x_{v_{n}}}=x_{\widehat{v}_{n}}\), is linear and continuous with respect to the convergence of the Boehmian spaces.

A similar proof for this theorem is available in various citations (see, e.g., [28] and [29]). Hence it is omitted.

We introduce the inverse operator of the operator \(x_{\widehat{v}_{n}}\) as follows.

Definition 21

Let \(x_{\widehat{v}_{n}}\in H^{S}\), \(x_{\widehat{v}_{n}}= ( \frac{\widehat{v}_{n}}{\widehat{w}_{n}} ) \) for \(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \). We define the inverse operator of \(x_{\widehat{v}_{n}}\) as

$$ \widehat{v}^{-1}x_{\widehat{v}_{n}}=x_{v_{n}}. $$
(13)

We derive the following inversion properties.

Theorem 22

Let\(x_{\widehat{v}_{n}}\in H^{S}\)and\(w=\widehat{v}\)for\(v\in L^{1}\). The inverse operator\(x_{\widehat{v}_{n}}\rightarrow x_{v_{n}}\)is linear. Moreover, it satisfies

$$ \widehat{v}^{-1} ( x_{\widehat{v}_{n}}\triangleright w ) =x_{v_{n}} \ast v \quad \textit{and}\quad \widehat{x_{v_{n}}\ast v}=x_{\widehat{v}_{n}} \triangleright w. $$

Proof

Consider two arbitrary Boehmians \(x_{\widehat{v}_{n}}\) and \(x_{\widehat{u}_{n}}\) in \(H^{S}\) defined, respectively, by \(x_{\widehat{v}_{n}}= ( \frac{\widehat{v}_{n}}{\widehat{w}_{n}} ) \) and \(x_{\widehat{u}_{n}}= ( \frac{\widehat{u}_{n}}{\widehat{\epsilon }_{n}} ) \). Then, for all \(n\in \mathbb{N}\), Theorem 2 and the linearity of the integral reveal

$$ x_{\widehat{v}_{n}}+x_{\widehat{u}_{n}}= \biggl( \frac{\widehat{v}_{n}\triangleright \widehat{\epsilon }_{n}+\widehat{u}_{n}\triangleright \widehat{w}_{n}}{\widehat{w}_{n}\triangleright \widehat{\epsilon }_{n}} \biggr) = \biggl( \frac{\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L} ( \widehat{v_{n}\ast \epsilon _{n}}+\widehat{u_{n}\ast w_{n}} ) }{\frac{\exp ( -i\pi x^{2}/\lambda d ) }{L}\widehat{ ( w_{n}\ast \epsilon _{n} ) }} \biggr) . $$

Hence, Definition 21 and the notion of addition in \(B^{S}\) imply

$$ \widehat{v}^{-1} ( x_{\widehat{v}_{n}}+x_{\widehat{u}_{n}} ) = \biggl( \frac{v_{n}\ast \epsilon _{n}+u_{n}\ast w_{n}}{w_{n}\ast \epsilon _{n}} \biggr) = \biggl( \frac{v_{n}}{w_{n}} \biggr) + \biggl( \frac{u_{n}}{\epsilon _{n}} \biggr) . $$

In notations, this gives

$$ \widehat{v}^{-1} ( x_{\widehat{v}_{n}}+x_{\widehat{u}_{n}} ) =x_{v_{n}}+x_{u_{n}}. $$

To complete the proof of linearity, indeed, for some \(\eta \in \mathbb{C}\) and all \(n\in \mathbb{N}\), we have

$$ \widehat{v}^{-1} ( \eta x_{\widehat{v}_{n}} ) =\eta \widehat{v}^{-1}x_{\widehat{v}_{n}}. $$

This proves the linearity condition. To show \(\widehat{v}^{-1} ( x_{\widehat{v}_{n}}\triangleright w ) =x_{v_{n}}\ast v\), we assume \(x_{v_{n}}= ( \frac{v_{n}}{w_{n}} ) \) in \(H^{S} \). Then, for every \(w\in L^{1}\), we have \(\widehat{v}^{-1} ( x_{\widehat{v}_{n}}\triangleright w ) = ( \frac{\widehat{v}_{n}\triangleright \widehat{v}}{\widehat{w}_{n}} ) \). By using Theorem 2 and Definition 21, we get

$$ \widehat{v}^{-1} ( x_{\widehat{v}_{n}}\triangleright w ) = \widehat{v}^{-1} \biggl( \frac{\widehat{v_{n}\ast v}}{\widehat{w}_{n}} \biggr) = \biggl( \frac{v_{n}}{w_{n}}\ast v \biggr) =x_{v_{n}}\ast v. $$

Proof of the part \(\widehat{x_{v_{n}}\ast v}=x_{\widehat{v}_{n}}\triangleright w\) is almost similar to the inversion case.

This finishes the proof of the theorem. □

5 Conclusion

In this article, the paraxial diffraction integral operator has been extended to a class of Boehmians, and the new operator has been obtained as a Boehmian. Many properties of the generalized diffraction integral operator coincide with the estimated properties of the classical operator. Various embeddings are also defined.