On Diffraction Fresnel Transforms for Boehmians

Abstract

The theory of the diffraction Fresnel transform is extended to certain spaces of Schwartz distributions. In the context of Boehmian spaces, the diffraction Fresnel transform is obtained as a continuous function. Convergence with respect to 𝛿 and Δ is also defined.

1. Introduction

The integral transforms play important role in the various fields of optics. One of great importance in many applications is the Fourier transform, where the kernel takes the form of a complex exponential function. The generalization of the Fourier transform is known as the fractional Fourier transform which was introduced by Namias in [1] and, has recently attracted considerable attention in optics and the light propagation in gradient-index media; see, for example, [2, 3], similarly in some lens systems see [4, 5]. Another well-known linear transform is the Fresnel transform; see [4–7], where the complex version of kernel having a quadratic combination of 𝑡 and 𝜉 in the exponent, see [8]. Recently, much attention has been paid to the diffraction Fresnel transform𝔉_𝑑𝑓(𝜉)=_𝐑𝐾𝛼₁,𝛾₁,𝛾₂,𝛼₂;𝜉,𝑡𝑓(𝑡)𝑑𝑡,(1.1) where𝐾𝛼₁,𝛾₁,𝛾₂,𝛼₂=1;𝜉,𝑡2𝜋𝑖𝛾₁𝑖exp2𝛾₁𝛼₁𝑡²−2𝜉𝑡+𝛼₂𝜉²(1.2) is the transform kernel with the real parameters and 𝛼₁,𝛾₁, and 𝛾₂ satisfy the following relation:𝛼₁𝛼₂−𝛾₁𝛾₂=1(1.3) holds; see [9].

Many familiar transforms can be considered as special cases of the generalized Fresnel transform. For example, if the parameters 𝛼₁,𝛾₁,𝛾₂ and 𝛼₂ satisfy the matrix𝛼₁𝛾₁𝛾₂𝛼₂=cos𝜃sin𝜃−sin𝜃cos𝜃(1.4) then the generalized Fresnel transform becomes a fractional Fourier transform.

In particular, when 𝜃=𝜋/2, one obtains the standard Fourier transform. Further, if 𝛼₁=𝛼₂=1, the generalized Fresnel transform reduces to the complex form of the Fresnel transform.

In the present paper, we show that the diffraction Fresnel transform can be extended to certain spaces generalized functions. In Section 2, we extend the diffraction Fresnel transform to a space of tempered distributions and further, by the aid of the Parseval’s equation, to a space of distributions of compact support. In Section 3, we define the diffraction Fresnel transform of a Boehmian and discuss its continuity with respect to 𝛿 and Δ convergence.

2. The Distributional Diffraction Fresnel Transform

Let 𝑆 denote the space of all complex valued functions 𝜙(𝑡) that are infinitely smooth and are such that, as |𝑡|→∞, they and their partial derivatives decrease to zero faster than every power of 1/|𝑡|. When 𝑡 is one dimensional, every function 𝜙(𝑡) in 𝑆 satisfies the infinite set of inequalities𝑡^𝑚𝜙^(𝑘)(𝑡)≤𝐶_𝑚,𝑘,where𝑡∈𝐑,(2.1) where 𝑚 and 𝑘 run through all nonnegative integers. The above expression can be interpreted aslim_|𝑡|→∞𝑡^𝑚𝜙^(𝑘)(𝑡)=0.(2.2) Members of 𝑆 are the so-called testing functions of rapid descent, then 𝑆 is naturally a linear space. The dual space 𝑆́ of 𝑆 is the space of distributions of slow growth (the space of tempered distributions). See [2, 10, 11].

Theorem 2.1. If 𝜙(𝑡) is in 𝑆, then its diffraction Fresnel transform 𝔉_𝑑𝜙𝜉)=1(2𝜋𝑖𝛾_1𝐑𝜙(𝑡)𝑖𝛼exp¹𝑡²−2𝑡𝜉+𝛼₂𝜉²2𝛾₁𝑑𝑡(2.3) exists and further also in 𝑆.

Proof. Let 𝜉 be fixed. If 𝜙(𝑡) is in 𝑆, then its diffraction Fresnel transform certainly exists. Moreover, differentiating the right-hand side of (2.3) with respect to 𝜉, under the integral sign, 𝑘-times, yields a sum of polynomials, 𝑝_𝑘(𝑡+𝜉), say of combinations of 𝑡 and 𝜉. That is, 𝑑^𝑘𝑑𝑡^𝑘𝔉_𝑑𝜙(𝜉)=𝑝_𝑘()𝜙(𝑡)𝑖𝛼𝑡+𝜉exp¹𝑡²−2𝑡𝜉+𝛼₂𝜉²2𝛾₁≤𝑝_𝑘(𝑡+𝜉)𝜙(𝑡),(2.4) which is also in 𝑆, since 𝜙 in 𝑆 and 𝑆 is a linear space. Hence, 𝜉^𝑚𝑑^𝑘𝑑𝑡^𝑘𝔉_𝑑𝜙(𝜉)≤_𝐑𝜉^𝑚𝑝_𝑘(𝑡+𝜉)𝜙(𝑡)𝑑𝑡.(2.5) Once again, since 𝜙∈𝑆, the integral on the right-hand side of (2.5) is bounded by a constant 𝐶_𝑚,𝑘, for every pair of nonnegative integers 𝑚 and 𝑘. Hence, we have the following theorem.

Theorem 2.2 (Parseval’s Equation for the diffraction transform). If 𝑓(𝑥) and 𝑔(𝑥) are absolutely integrable, over 𝑥∈𝐑, then _𝐑𝑓(𝑥)𝔉_𝑑𝑔(𝑥)𝑑𝑥=_𝐑𝔉_𝑑𝑓(𝑥)𝑔(𝑥)𝑑𝑥,(2.6) where 𝔉_𝑑𝑓 and𝔉_𝑑𝑔 are the corresponding diffraction Fresnel transforms of 𝑓 and 𝑔, respectively.

Proof. The diffraction Fresnel transforms 𝔉_𝑑𝑓(𝜉) and 𝔉_𝑑𝑔 (𝜉) are indeed bounded and continuous for all 𝜉. This ensure the convergence of the integrals in (2.6). Moreover, _𝐑𝑓(𝑥)𝔉_𝑑𝑔(𝑥)𝑑𝑥=_𝐑𝑑𝑥_𝐑𝑓(𝑥)𝑔𝑦𝑖𝛼exp¹𝑦²−2𝑥𝑦+𝛼₂𝑥²2𝛾₁𝑑𝑦,𝛼₁𝛼₂−𝛾₁𝛾₂=1.(2.7) Since the integral (2.7) is absolutely integrable over the entire (𝑥,𝑦)-plane, Fubini’s theorem allows us to interchange the order of integration. Hence, (2.7) can be written as _𝐑𝑓(𝑥)_𝐑𝑔𝑦𝑖𝛼exp²𝑥²−2𝑥𝑦+𝛼₁𝑦²2𝛾₁𝑑𝑦𝑑𝑥=_𝐑𝔉_𝑑𝑓𝑦𝑔𝑦𝑑𝑦,(2.8) where 𝛼₂𝛼₁−𝛾₁𝛾₂=1. This completes the proof of the theorem.

Parseval’s relation can be interpreted as𝔉_𝑑𝑓,𝜙=𝑓,𝔉_𝑑𝜙.(2.9) Therefore, from the above relation, we state the diffraction Fresnel transform of a distribution 𝑓 of slow growth (𝑓∈𝑆)́ as𝔉_𝑑𝑓,𝜙=𝑓,𝔉_𝑑𝜙,∀𝜙∈𝑆,(2.10) and it is well defined by Theorem 2.1.

Theorem 2.3. If 𝑓 is a distribution of slow growth, then its diffraction Fresnel transform 𝔉_𝑑𝑓 is also a distribution of slow growth.

Proof. Linearity of 𝔉_𝑑𝑓 is obvious. To show continuity of 𝔉_𝑑𝑓, let (𝜙_𝑛)^∞_𝑛=1→0, in 𝑆, then also (𝔉_𝑑𝜙_𝑛)^∞_𝑛=1→0 in 𝑆 as 𝑛→∞. Hence, 𝔉_𝑑𝑓,𝜙_𝑛=𝑓,𝔉_𝑑𝜙_𝑛⟶0as𝑛⟶∞.(2.11) Hence 𝔉_𝑑𝑓∈𝑆́. This completes the proof of the theorem.

Theorem 2.4. Let 𝑓 be a distribution of compact support (𝑓∈𝐸)́. Then, we define the Fresnel transform of 𝑓 as 𝔉_𝑑𝑓(𝜉)=12𝜋𝑖𝛾₁𝑓(𝑡)𝑖𝛼,exp¹𝑡²−2𝑡𝜉+𝛼₂𝜉²2𝛾₁.(2.12)

Proof. Let 𝜙∈𝑆(ℝ) be arbitrary. From (2.10), we read 𝔉_𝑑𝑓(𝜉),𝜙(𝜉)=𝑓(𝑡),𝔉_𝑑𝜙(𝑡=1)2𝜋𝑖𝛾₁𝑓(𝑡),_𝐑𝜙(𝜉)𝑖𝛼exp¹𝜉²−2𝑡𝜉+𝛼₂𝑡²2𝛾₁=1𝑑𝜉2𝜋𝑖𝛾_1𝐑𝑓(𝑡)𝑖𝛼,exp²𝑡²+−2𝑡𝜉+𝛼₁𝜉²2𝛾₁𝜙(𝜉)=1𝑑𝜉2𝜋𝑖𝛾₁𝑓(𝑡)𝑖𝛼,exp²𝑡²+−2𝑡𝜉+𝛼₁𝜉²2𝛾₁,𝜙(𝜉).(2.13) But since ⟨𝑓(𝑡),exp(𝑖(𝛼₂𝑡²+−2𝑡𝜉+𝛼₁𝜉²)/2𝛾₁)⟩ is an infinitely smooth function, we get 𝔉_𝑑𝑓(𝜉)=12𝜋𝑖𝛾₁𝑓(𝑡)𝑖𝛼,exp²𝑡²+−2𝑡𝜉+𝛼₁𝜉²2𝛾₁.(2.14) This completes the proof of the theorem.

Now, for distributions 𝑓 and 𝑔∈𝐸(́𝐑), we define the convolution product as 𝑓∗9(𝑡),𝜙(𝑡)=𝑓(𝑡),𝑔(𝜏),𝜙(𝑡+𝜏),(2.15) for every 𝜙∈𝐸(𝐑). This definition makes sense, since ⟨𝑔(𝜏),𝜙(𝑡+𝜏)⟩ belongs to 𝔇, and hence a member of 𝐸(𝐑). With this definition, we are allowed to write the following theorem.

Theorem 2.5. For every 𝑓∈𝐸(́𝐑), the function 𝜓(𝑡)=⟨𝑓(𝜏),𝜙(𝑡+𝜏)⟩  is infinitely smooth and satisfies the relation 𝐷^𝑘_𝑡𝜓(𝑡)=𝑓(𝜏),𝐷^𝑘_𝑡𝜙(𝑡+𝜏),(2.16) for all 𝑘∈𝐍.

Proof (see page 26 in [12]) . A direct result of the convolution product is the following theorem.

Theorem 2.6 (Convolution Theorem). Let 𝑓 and 𝑔 be distributions of compact support and 𝔉_𝑑𝑓(𝜉)=𝔉_𝑑(𝑓(𝑡);𝜉),𝔉_𝑑𝑔(𝜉)=𝔉_𝑑(𝑔(𝜏);𝜉) their respective diffraction Fresnel transforms, then𝔉_𝑑𝑓∗𝑔(𝑡);𝜉=2𝜋𝑖𝛾₁𝑖exp2𝛼₁𝑡𝜏−𝛼₂𝜉²2𝛾₁𝔉_𝑑𝑓(𝑡);𝜉𝔉_𝑑𝑔(𝜏);𝜉.(2.17)

Proof. Let 𝑓,𝑔∈𝐸(́𝐑), then by using (2.12), we get 𝔉_𝑑𝑡)=1𝑓∗𝑔(;𝜉2𝜋𝑖𝛾₁𝑡)𝑖𝛼𝑓∗𝑔(,exp¹𝑡²−2𝑡𝜉+𝛼₂𝜉²2𝛾₁1i.e.=2𝜋𝑖𝛾₁𝑓(𝑡),𝑔(𝜏)𝑖𝛼,exp¹(𝑡+𝜏)²−(𝑡+𝜏)𝜉+𝛼₂𝜉²2𝛾₁=12𝜋𝑖𝛾₁𝑓(𝑡),𝑔(𝜏)𝑖𝛼,exp¹𝑡²+𝛼₁𝜏²+𝛼₁𝑡𝜏−2𝑡𝜉−2𝜏𝜉+𝛼₂𝜉²2𝛾₁.(2.18) Properties of distributions together with simple calculations on the exponent yield 𝔉_𝑑𝑓∗𝑔(𝑡);𝜉=2𝜋𝑖𝛾₁𝑖exp2𝛼₁𝑡𝜏−𝛼₂𝜉²2𝛾₁𝔉_𝑑𝑓(𝑡);𝜉𝔉_𝑑𝑔(𝜏);𝜉.(2.19) This completes the proof of the theorem.

Corollary 2.7. Let 𝑓,𝑔∈𝐸(́𝐑), then (1)𝔉_𝑑𝑓∗𝛿_𝑛(𝑡);𝜉=2𝜋𝑖𝛾₁−𝛼exp²𝜉²2𝛾₁𝔉_𝑑𝑓(𝜉),(2)𝔉_𝑑𝛿_𝑛∗𝑔(𝑡);𝜉=2𝜋𝑖𝛾₁−𝛼exp²𝜉²2𝛾₁𝔉_𝑑𝑔(𝜉),(2.20) where 𝔉_𝑑𝑓(𝜉)=𝔉_𝑑(𝑓(𝑡);𝜉),𝔉_𝑑𝑔(𝜉)=𝔉_𝑑(𝑔(𝜏);𝜉).
The following is a theorem which can be directly established from (2.12) and the fact that [11] 𝐷^𝑘𝑓∗𝑔=𝐷^𝑘𝑓∗𝑔=𝑓∗𝐷^𝑘𝑔.(2.21)

Theorem 2.8. Let 𝑓 and 𝑔 be distributions of compact support and 𝔉_𝑑𝑓(𝜉)=𝔉_𝑑(𝑓(𝑡);𝜉),𝔉_𝑑𝑔(𝜉)=𝔉_𝑑(𝑔(𝜏);𝜉) their respective diffraction Fresnel transforms, then (1)𝔉_𝑑𝐷^𝑘_𝑡𝑓∗𝑔(𝑡);𝜉=2𝜋𝑖𝛾₁𝑖exp2𝛼₁𝑡𝜏−𝛼₂𝜉²2𝛾₁𝔉_𝑑𝑓^(𝑘)(𝑡);𝜉𝔉_𝑑𝑔(𝜉),(2)𝔉_𝑑𝐷^𝑘_𝑡𝑓∗𝑔(𝑡);𝜉=2𝜋𝑖𝛾₁𝑖exp2𝛼₁𝑡𝜏−𝛼₂𝜉²2𝛾₁𝔉_𝑑𝑓(𝜉)𝔉_𝑑𝑔^(𝑘)(𝜏);𝜉.(2.22)

3. Diffraction Fresnel Transform of Boehmians

Let 𝔛 be a linear space and ℑ a subspace of 𝔛. To each pair of elements 𝑓∈𝔛 and 𝜙∈ℑ, we assign a product 𝑓⋅𝑔 such that the following conditions are satisfied:(i)if 𝜙,𝜓∈ℑ, then 𝜙⋅𝜓∈ℑ and 𝜙⋅𝜓=𝜓⋅𝜙,(ii)if 𝑓∈𝔛 and 𝜙,𝜓∈ℑ, then (𝑓⋅𝜙)⋅𝜓=𝑓⋅(𝜙⋅𝜓),(iii)if 𝑓,𝑔∈𝔛,𝜙∈ℑ and 𝜆∈𝐑, then (𝑓+𝑔)⋅𝜙=𝑓⋅𝜙+𝑔⋅𝜙 and 𝜆(𝑓⋅𝜙)=(𝜆𝑓)⋅𝜙. Let Δ be a family of sequences from ℑ such that(a)if 𝑓,𝑔∈𝔛,(𝛿_𝑛)∈Δ and 𝑓⋅𝛿_𝑛=𝑔⋅𝛿_𝑛(𝑛=1,2,…), then 𝑓=𝑔,(b)if (𝜙_𝑛),(𝛿_𝑛)∈Δ, then (𝜙_𝑛⋅𝜓_𝑛)∈Δ.

Elements of Δ will be called delta sequences. Consider the class 𝐔 of pair of sequences defined by𝐔=𝑓_𝑛,𝜙_𝑛∶𝑓_𝑛⊆𝔛^𝐍,𝜙_𝑛∈Δ,(3.1) for each 𝑛∈𝐍. An element ((𝑓_𝑛),(𝜙_𝑛))∈𝐔 is called a quotient of sequences, denoted by 𝑓_𝑛/𝜙_𝑛, or [𝑓_𝑛/𝜙_𝑛] if 𝑓_𝑖⋅𝜙_𝑗=𝑓_𝑗⋅𝜙_𝑖,forall𝑖,𝑗∈𝐍.

Similarly, two quotients of sequences 𝑓_𝑛/𝜙_𝑛 and 𝑔_𝑛/𝜓_𝑛 are said to be equivalent, 𝑓_𝑛/𝜙_𝑛∼𝑔_𝑛/𝜓_𝑛, if 𝑓_𝑖⋅𝜓_𝑗=𝑔_𝑗⋅𝜙_𝑖,forall𝑖,𝑗∈𝐍. The relation ~ is an equivalent relation on 𝐔, and hence splits 𝐔 into equivalence classes. The equivalence class containing 𝑓_𝑛/𝜙_𝑛 is denoted by [𝑓_𝑛/𝜙_𝑛]. These equivalence classes are called Boehmians, and the space of all Boehmians is denoted by 𝔅.

The sum of two Boehmians and multiplication by a scalar can be defined in a natural way𝑓_𝑛𝜙_𝑛+𝑔_𝑛𝜓_𝑛=𝑓_𝑛⋅𝜓_𝑛+𝑔_𝑛⋅𝜙_𝑛𝜙_𝑛⋅𝜓_𝑛,𝛼𝑓_𝑛𝜙_𝑛=𝛼𝑓_𝑛𝜙_𝑛,𝛼∈ℂ.(3.2) The operation ⋅ and the differentiation are defined by𝑓_𝑛𝜙_𝑛⋅𝑔_𝑛𝜓_𝑛=𝑓_𝑛⋅𝑔_𝑛𝜙_𝑛⋅𝜓_𝑛,𝒟^𝛼𝑓_𝑛𝜙_𝑛=𝒟^𝛼𝑓_𝑛𝜙_𝑛.(3.3) The relationship between the notion of convergence and the product ⋅ are given by the following:(i)if 𝑓_𝑛→𝑓 as 𝑛→∞ in 𝔛 and, 𝜙∈ℑ is any fixed element, then 𝑓_𝑛⋅𝜙→𝑓⋅𝜙 in 𝔛(as𝑛→∞),(ii)if 𝑓_𝑛→𝑓 as 𝑛→∞ in 𝔛 and (𝛿_𝑛)∈Δ, then 𝑓_𝑛⋅𝛿_𝑛→𝑓 in 𝔛(as𝑛→∞).

The operation ⋅ can be extended to 𝔅×ℑ by𝑓If^𝑛𝛿_𝑛𝑓∈𝔅and𝜙∈ℑ,then^𝑛𝛿_𝑛𝑓⋅𝜙=^𝑛⋅𝜙𝛿_𝑛.(3.4) In 𝔅, one can define two types of convergence as follows:(i)(𝛿-convergence) a sequence (𝛽_𝑛) in 𝔅 is said to be 𝛿-convergent to 𝛽 in 𝔅, denoted by 𝛽→−_𝑛^𝛿𝛽, if there exists a delta sequence (𝛿_𝑛) such that (𝛽_𝑛⋅𝛿_𝑛),(𝛽⋅𝛿_𝑛)∈𝔛,forall𝑘,𝑛∈𝐍,   and (𝛽_𝑛⋅𝛿_𝑘)→(𝛽⋅𝛿_𝑘)  as 𝑛→∞,  in 𝔛,   for every 𝑘∈𝐍,(ii)(Δ-convergence) a sequence (𝛽_𝑛) in 𝔅is said to be Δ-convergent to 𝛽 in 𝔅, denoted by 𝛽−→_𝑛^Δ𝛽, if there exists a (𝛿_𝑛)∈Δ such that (𝛽_𝑛−𝛽)⋅𝛿_𝑛∈𝔛,forall𝑛∈𝐍, and (𝛽_𝑛−𝛽)⋅𝛿_𝑛→0 as 𝑛→∞ in 𝔛.

For further analysis we refer, for example, to [10, 13–19]. Now we let 𝐿¹ be the space of Lebesgue integrable functions on 𝐑 and 𝔅_𝐿¹ the space of Lebesgue integrable Boehmians [17] with the set Δ of all delta sequence (𝛿_𝑛) from 𝔇 (the test function space of compact support) such that(1)∫_𝐑𝛿_𝑛=1 for all 𝑛∈𝐍,(2)∫_𝐑|𝛿_𝑛|<𝑀 for certain positive number 𝑀 and 𝑛∈𝐍,(3)∫_|𝑡|>𝜀|𝛿_𝑛(𝑡)|𝑑𝑡→0 as 𝑛→∞ for every 𝜀>0.

Then, 𝔅_𝐿¹ is a convolution algebra with the pointwise operations(i)𝜆[𝑓_𝑛/𝛿_𝑛]=[𝜆𝑓_𝑛/𝛿_𝑛], (ii)[𝑓_𝑛/𝛿_𝑛]+[𝑔_𝑛/𝜙_𝑛]=[(𝑓_𝑛∗𝜙_𝑛+𝑔_𝑛∗𝛿_𝑛)/(𝛿_𝑛∗𝜙_𝑛)], (iii)and the convolution 𝑓_𝑛𝛿_𝑛∗𝑔_𝑛𝜙_𝑛=𝑓_𝑛∗𝑔_𝑛𝛿_𝑛∗𝜙_𝑛.(3.5)

Lemma 3.1. Let [𝑓_𝑛/𝛿_𝑛]∈𝔅_𝐿¹, then the sequence 𝔉_𝑑𝑓_𝑛(𝑡)=1;𝜉2𝜋𝑖𝛾_1𝐑𝑓_𝑛(𝑡)𝑖exp2𝛾₁𝛼₁𝑡²−2𝑡𝜉+𝛼₂𝜉²𝑑𝑡(3.6) converges uniformly on each compact set 𝐾 in 𝐑.

Proof. Let 𝑓_𝑛=𝔉_𝑑𝑓. For each compact set 𝐾,𝛿_𝑛(𝛿_𝑛=𝔉_𝑑𝛿_𝑛) converges uniformly to the function exp(−(𝑖𝛼₂/2𝛾₁)𝜉²). Hence, by Corollary 2.7, 𝔉_𝑑𝑓_𝑛(𝑡);𝜉=𝑓_𝑛𝛿_𝑘𝛿_𝑘=𝑒^{(𝑖𝛼2/2𝛾1)𝜉2}2𝜋𝑖𝛾₁𝔉_𝑑𝑓_𝑛∗𝛿_𝑘𝛿_𝑘.(3.7) Using the choice 𝑓_𝑛/𝛿_𝑛 that is quotient of sequences and upon employing Corollary 2.7, we have 𝔉_𝑑𝑓_𝑛(𝑡)=𝑒;𝜉^{(𝑖𝛼2/2𝛾1)𝜉2}2𝜋𝑖𝛾₁𝔉_𝑑𝑓_𝑘∗𝛿_𝑛𝛿_𝑘=𝑓_𝑘𝛿_𝑘𝛿_𝑛=𝑓_𝑘𝛿_𝑘2𝜋𝑖𝛾₁𝑒^{−(𝑖𝛼2/2𝛾1)𝜉2}.(3.8) This completes the proof of the Lemma.

By using this Lemma, we are able to define the diffractional Fresnel transform of a Boehmian as follows: [𝑓_𝑛/𝛿_𝑛]  in 𝔅_𝐿¹ as ℛ𝑓_𝑛𝛿_𝑛=lim_𝑛→∞𝑓_𝑛,(3.9)where the limit ranges over compact subsets of 𝐑. Now, let [𝑋_𝑛/𝛿_𝑛]=[𝑌_𝑛/𝛾_𝑛] in 𝔅_𝐿¹, then𝑋_𝑛∗𝛾_𝑚=𝑌_𝑚∗𝛿_𝑛,forevery𝑚,𝑛∈𝐍.(3.10) Hence, employing the Fresnel transform to both sides of above equation implies𝔉_𝑑𝑋_𝑛∗𝛾_𝑚=𝔉_𝑑𝑌_𝑚∗𝛿_𝑛=𝔉_𝑑𝑌_𝑛∗𝛿_𝑚.(3.11) Thus, using Theorem 2.6 and the fact that𝛿_𝑛and𝛿_𝑚⟶2𝜋𝑖𝛾₁𝑒^{−(𝑖𝛼2/2𝛾1)𝜉2},(3.12) on compact subsets of 𝐑, we getlim_𝑛→∞𝔉_𝑑𝑋_𝑛=lim_𝑛→∞𝔉_𝑑𝑌_𝑛.(3.13) Hence,ℛ𝑋_𝑛𝛿_𝑛𝑌=ℛ^𝑛𝛾_𝑛.(3.14) The definition is therefore well defined.

Theorem 3.2. Let 𝐵₁ and 𝐵₂ be in 𝔅_𝐿¹ and 𝛼∈ℂ, then(i)ℛ(𝛼𝐵₁)=𝛼ℛ𝐵₁, (ii)ℛ(𝐵₁+𝐵₂)=ℛ𝐵₁+ℛ𝐵₂,(iii)ℛ(𝐵₁∗𝛿_𝑛)=2𝜋𝑖𝛾₁𝑒^{−(𝑖𝛼2/2𝛾1)𝜉2}ℛ𝐵₁=ℛ(𝛿_𝑛∗𝐵₁),(iv) if ℛ𝐵₁=0, then 𝐵₁=0,(v)if 𝐵−→_𝑛^Δ𝐵 as 𝑛→∞ in ℬ_𝐿¹, then ℛ𝐵−→_𝑛^Δℛ𝐵 as 𝑛→∞ in 𝔅_𝐿¹ on compact subsets.

Proof. The proof of (i), (ii), and (iv) follows from the corresponding properties of the distributional Fresnel transform. Since each 𝑓∈𝐸́ has a representative 𝑓⟶𝑓∗𝜙_𝑛𝜙_𝑛,(3.15) in the space 𝔅_𝐿¹, Part (iii) follows from Corollary  2.7. Finally, the proof of Part (v) is analogous to that employed for the proof of Part (f) of [17, Theorem  2]. This completes the proof of the theorem.

Theorem 3.3. The Fresnel transform ℛ is continuous with respect to the 𝛿-convergence.

Proof. Let 𝐵→−_𝑛^𝛿𝐵 in 𝐵_𝐿¹ as 𝑛→∞, then we show that ℛ𝐵→−_𝑛^𝛿ℛ𝐵 as 𝑛→∞. Using [17, Theorem  2.6], we find [𝑓_𝑛,𝑘/𝛿_𝑘]=𝐵_𝑛 and [𝑓_𝑘/𝛿_𝑘]=𝐵 such that 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞,𝑘∈𝐍. Applying the Fresnel transform for both sides implies 𝑓_𝑛,𝑘→𝑓_𝑘 in the space of continuous functions. Therefore, considering limits, we get ℛ𝑓_𝑛,𝑘𝛿_𝑘𝑓⟶ℛ^𝑘𝛿_𝑘.(3.16) This completes the proof of the theorem.

Theorem 3.4. The diffraction Fresnel transform ℛ  is continuous with respect to the Δ-convergence.

Proof. Let 𝐵−→_𝑛^Δ𝐵 as 𝑛→∞ in 𝑛→∞, then there is 𝑓_𝑛∈𝐿¹ and 𝛿_𝑛∈Δ such that 𝐵_𝑛−𝐵∗𝛿_𝑛=𝑓_𝑛∗𝛿_𝑛𝛿_𝑘,𝑓_𝑛→0as𝑛→∞.(3.17) Thus ℛ𝐵_𝑛−𝐵∗𝛿_𝑛𝑓=ℛ^𝑛∗𝛿_𝑛𝛿_𝑘⟶𝔉_𝑑𝑓_𝑛∗𝛿_𝑛⟶as𝑛→∞2𝜋𝑖𝛾₁𝑒^{−(𝑖𝛼2/2𝛾1)𝜉2}𝔉_𝑑𝑓_𝑛as𝑛⟶∞byCorollary2.7⟶0bythelinearityof𝔉_𝑑𝑓_𝑛.(3.18) Therefore, ℛ(𝐵_𝑛−𝐵)→0 as 𝑛→∞. Thus, ℛ𝐵−→_𝑛^Δℛ𝐵 as 𝑛→∞. This completes the proof.

Lemma 3.5. Let [𝑓_𝑛/𝜙_𝑛]∈ℬ_𝐿¹ and 𝜙∈𝔇(𝐑), then ℛ𝑓_𝑛𝜙_𝑛∗𝜙=2𝜋𝑖𝛾₁𝑒^{𝑖(2𝛼1𝑡𝜏−𝛼2𝜉2)/2𝛾1}ℛ𝑓_𝑛𝜙_𝑛∗𝔉_𝑑𝜙.(3.19)

Proof. Let [𝑓_𝑛/𝜙_𝑛]∈𝔅_𝐿¹, then using (3.9), we have ℛ𝑓_𝑛𝜙_𝑛𝑓∗𝜙=ℛ^𝑛∗𝜙𝜙_𝑛=lim_𝑛→∞𝔉_𝑑𝑓_𝑛∗𝜙,(3.20) on compact subsets of 𝐑. By applying Theorem 2.6, it yields ℛ𝑓_𝑛𝜙_𝑛∗𝜙=2𝜋𝑖𝛾₁𝑒^{𝑖(2𝛼1𝑡𝜏−𝛼2𝜉2)/2𝛾1}lim_𝑛→∞𝔉_𝑑𝑓(𝑡);𝜉𝔉_𝑑𝜙(𝜏);𝜉.(3.21) Hence, ℛ([𝑓_𝑛/𝜙_𝑛]∗𝜙)=2𝜋𝑖𝛾₁𝑒^{𝑖(2𝛼1𝑡𝜏−𝛼2𝜉2)/2𝛾1}ℛ[𝑓_𝑛/𝜙_𝑛]𝔉_𝑑(𝜙(𝜏);𝜉). This completes the proof of the lemma.

Acknowledgments

The authors would like to thank the referee for valuable remarks and suggestions on the previous version of the paper. The second author gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme  no. 05-01-09-0720RU.