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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

Optical integral transforms have been studied in several works, for example, [

The parameters

Note that many familiar transforms can be considered as special cases of the diffraction Fresnel transform. For example, if the parameters

In the present work, we consider a combined optical transform of Fresnel and wavelet transforms, namely, the optical Fresnel-wavelet transform defined by [

The parameters

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 [

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

Let

Further,

Let

where

Let

Let

Now

Let

It is a straightforward result of Lemma

The distributional optical Fresnel-wavelet transform

It is obvious.

Let

It is a straightforward conclusion of the fact [

In this section, we assume that the reader is acquainted with the general construction of Boehmian spaces [

A sequence

The set of all such sequences is denoted by

Given

We prove the lemma by induction on

Next, assume that the lemma satisfies for

Let

Let

The inequality (

Let

In view of Lemma

Therefore, the righthand side of (

Hence we have

Let

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

Let

Using (

Let

It is a straightforward result of definitions.

Let

By virtue of Lemma

Allowing

Let

Considering a compact subset

Then the mean value theorem implies that

Now allowing

On using (

This implies that

Finally, by virtue of the above sequence of results (Lemma

In this section we construct the space of all Fresnel-wavelet transforms of Boehmians from the space

Let

Let

By the aid of (

By using (

That is

Let

Let

Thus

Let

Theorem

Let

It is obvious.

(i) Let

(ii) Let

(i) Let

By Lemma

Once again Theorem

This proves part (i) of the Lemma.

(ii) can be proved similarly by using Lemma

The sum of two Boehmians and multiplication by a scalar in

The operation

In view of the analysis obtained in Sections

Let

The optical Fresnel-wavelet transform

It is a straightforward.

The optical Fresnel-wavelet transform

It is straightforward by using Definition

The optical Fresnel-wavelet transform

Assume that

Now, Let

Let

Applying Definition

Using Lemma

This completes the proof of the Lemma.

First of all, we show that

Let

such that

To prove the second part, let

Now, we establish continuity of

Hence, from Lemma

Hence,

Lemma

Thus we have

From this we find that

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.