# On Diffraction Fresnel Transforms for Boehmians

Authors: S. K. Q. Al-Omari; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2011)

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## 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 toδ and Δ is also defined.


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## Body

## 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 transform(1.1)Fdf(ξ)=RK(α1,γ1,γ2,α2;ξ,t)f(t)dt,
where(1.2)K(α1,γ1,γ2,α2;ξ,t)=12πiγ1exp⁡(i2γ1(α1t2-2ξt+α2ξ2))
is the transform kernel with the real parameters and α1,γ1, and γ2 satisfy the following relation:(1.3)α1α2-γ1γ2=1
holds; see [9].Many familiar transforms can be considered as special cases of the generalized Fresnel transform. For example, if the parametersα1,γ1,γ2 and α2 satisfy the matrix(1.4)(α1γ1γ2α2)=(cos⁡θsin⁡θ-sin⁡θcos⁡θ)
then the generalized Fresnel transform becomes a fractional Fourier transform.In particular, whenθ=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 Section2, 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

LetS 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 inequalities(2.1)|tmϕ(k)(t)|≤Cm,k,wheret∈R,
where m and k run through all nonnegative integers. The above expression can be interpreted as(2.2)lim⁡|t|→∞tmϕ(k)(t)=0.
Members of S are the so-called testing functions of rapid descent, then S is naturally a linear space. The dual space S´ of S is the space of distributions of slow growth (the space of tempered distributions). See [2, 10, 11].Theorem 2.1.
Ifϕ(t) is in  S, then its diffraction Fresnel transform
(2.3)Fd(ϕ)(ξ)=12πiγ1∫Rϕ(t)exp⁡(i(α1t2-2tξ+α2ξ2)2γ1)dt
exists and further also in  S.Proof.
Letξ 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, pk(t+ξ), say of combinations of t and ξ. That is,
(2.4)|dkdtkFd(ϕ)(ξ)|=|pk(t+ξ)ϕ(t)exp⁡(i(α1t2-2tξ+α2ξ2)2γ1)|≤|pk(t+ξ)ϕ(t)|,
which is also in S, since ϕ in S and S is a linear space. Hence,
(2.5)|ξmdkdtkFd(ϕ)(ξ)|≤∫R|ξmpk(t+ξ)ϕ(t)|dt.
Once again, since ϕ∈S, the integral on the right-hand side of (2.5) is bounded by a constant Cm,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).
Iff(x) and  g(x) are absolutely integrable, over  x∈R, then
(2.6)∫Rf(x)Fdg(x)dx=∫RFdf(x)g(x)dx,
where 𝔉df and𝔉dg are the corresponding diffraction Fresnel transforms of  f and  g, respectively.Proof.
The diffraction Fresnel transforms𝔉df(ξ) and 𝔉dg(ξ) are indeed bounded and continuous for all ξ. This ensure the convergence of the integrals in (2.6). Moreover,
(2.7)∫Rf(x)Fdg(x)dx=∫RdxRf(x)g(y)exp⁡⁡(i(α1y2-2xy+α2x2)2γ1)dy,α1α2-γ1γ2=1.
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
(2.8)∫Rf(x)Rg(y)exp⁡(i(α2x2-2xy+α1y2)2γ1)dydx=∫RFdf(y)g(y)dy,
where α2α1-γ1γ2=1. This completes the proof of the theorem.Parseval’s relation can be interpreted as(2.9)〈Fdf,ϕ〉=〈f,Fdϕ〉.
Therefore, from the above relation, we state the diffraction  Fresnel transform of a distribution f of slow growth (f∈S´) as(2.10)〈Fdf,ϕ〉=〈f,Fdϕ〉,∀ϕ∈S,
and it is well defined by Theorem 2.1.Theorem 2.3.
Iff is a distribution of slow growth, then its diffraction Fresnel transform  𝔉df is also a distribution of slow growth.Proof.
Linearity of𝔉df is obvious. To show continuity of 𝔉df, let (ϕn)n=1∞→0, in S, then also (𝔉dϕn)n=1∞→0 in S as n→∞. Hence,
(2.11)〈Fdf,ϕn〉=〈f,Fdϕn〉⟶0asn⟶∞.
Hence 𝔉df∈S´. This completes the proof of the theorem.Theorem 2.4.
Letf be a distribution of compact support  (f∈E´). Then, we define the Fresnel transform of  f as
(2.12)Fdf(ξ)=12πiγ1〈f(t),exp⁡(i(α1t2-2tξ+α2ξ2)2γ1)〉.Proof.
Letϕ∈S(ℝ) be arbitrary. From (2.10), we read
(2.13)〈Fdf(ξ),ϕ(ξ)〉=〈f(t),Fdϕ(t)〉=12πiγ1〈f(t),∫Rϕ(ξ)exp⁡(i(α1ξ2-2tξ+α2t2)2γ1)dξ〉=12πiγ1∫R〈f(t),exp⁡(i(α2t2+-2tξ+α1ξ2)2γ1)〉ϕ(ξ)dξ=12πiγ1〈〈f(t),exp⁡(i(α2t2+-2tξ+α1ξ2)2γ1)〉,ϕ(ξ)〉.
But since 〈f(t),exp(i(α2t2+-2tξ+α1ξ2)/2γ1)〉 is an infinitely smooth function, we get
(2.14)Fdf(ξ)=12πiγ1〈f(t),exp⁡(i(α2t2+-2tξ+α1ξ2)2γ1)〉.
This completes the proof of the theorem.Now, for distributionsf and g∈E´(R), we define the convolution product as (2.15)〈(f*9)(t),ϕ(t)〉=〈f(t),〈g(τ),ϕ(t+τ)〉〉,
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 everyf∈E´(R), the function ψ(t)=〈f(τ),ϕ(t+τ)〉  is infinitely smooth and satisfies the relation
(2.16)Dtkψ(t)=〈f(τ),Dtkϕ(t+τ)〉,
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).
Letf and  g be distributions of compact support and  𝔉df(ξ)=𝔉d(f(t);ξ),𝔉dg(ξ)=𝔉d(g(τ);ξ) their respective diffraction Fresnel transforms, then(2.17)Fd((f*g)(t);ξ)=2πiγ1exp⁡(i(2α1tτ-α2ξ2)2γ1)Fd(f(t);ξ)Fd(g(τ);ξ).Proof.
Letf,g∈E´(R), then by using (2.12), we get
(2.18)Fd((f*g)(t);ξ)=12πiγ1〈(f*g)(t),exp⁡(i(α1t2-2tξ+α2ξ2)2γ1)〉i.e.=12πiγ1〈f(t),〈g(τ),exp⁡(i(α1(t+τ)2-(t+τ)ξ+α2ξ2)2γ1)〉〉=12πiγ1〈f(t),〈g(τ),exp⁡(i(α1t2+α1τ2+α1tτ-2tξ-2τξ+α2ξ2)2γ1)〉〉.
Properties of distributions together with simple calculations on the exponent yield
(2.19)Fd((f*g)(t);ξ)=2πiγ1exp⁡(i(2α1tτ-α2ξ2)2γ1)Fd(f(t);ξ)Fd(g(τ);ξ).
This completes the proof of the theorem.Corollary 2.7.
Letf,g∈E´(R), then
(2.20)(1)Fd(f*δn(t);ξ)=2πiγ1exp⁡(-α2ξ22γ1)Fd(f)(ξ),(2)Fd(δn*g(t);ξ)=2πiγ1exp⁡(-α2ξ22γ1)Fd(g)(ξ),
where 𝔉df(ξ)=𝔉d(f(t);ξ),𝔉dg(ξ)=𝔉d(g(τ);ξ).
The following is a theorem which can be directly established from (2.12) and the fact that [11]
(2.21)Dk(f*g)=Dkf*g=f*Dkg.Theorem 2.8.
Letf and  g be distributions of compact support and  𝔉df(ξ)=𝔉d(f(t);ξ),𝔉dg(ξ)=𝔉d(g(τ);ξ) their respective diffraction Fresnel transforms, then
(2.22)(1)Fd(Dtk(f*g)(t);ξ)=2πiγ1exp⁡(i(2α1tτ-α2ξ2)2γ1)Fd(f(k)(t);ξ)Fdg(ξ),(2)Fd(Dtk(f*g)(t);ξ)=2πiγ1exp⁡(i(2α1tτ-α2ξ2)2γ1)Fdf(ξ)Fd(g(k)(τ);ξ).


## 3. Diffraction Fresnel Transform of Boehmians

Let𝔛 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)
ifϕ,ψ∈ℑ, then ϕ·ψ∈ℑ and ϕ·ψ=ψ·ϕ,(ii)
iff∈𝔛 and ϕ,ψ∈ℑ, then (f·ϕ)·ψ=f·(ϕ·ψ),(iii)
iff,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)
iff,g∈𝔛,(δn)∈Δ and f·δn=g·δn(n=1,2,…), then f=g,(b)
if(ϕn),(δn)∈Δ, then (ϕn·ψn)∈Δ.Elements ofΔ will be called delta sequences.Consider the class U of pair of sequences defined by(3.1)U={((fn),(ϕn)):(fn)⊆XN,(ϕn)∈Δ},
for each n∈N. An element ((fn),(ϕn))∈U is called a quotient of sequences, denoted by fn/ϕn, or [fn/ϕn] if fi·ϕj=fj·ϕi,foralli,j∈N.Similarly, two quotients of sequencesfn/ϕn and gn/ψn are said to be equivalent, fn/ϕn~gn/ψn, if fi·ψj=gj·ϕi,foralli,j∈N. The relation ~ is an equivalent relation on U, and hence splits U into equivalence classes. The equivalence class containing fn/ϕn is denoted by [fn/ϕn]. These equivalence classes are called Boehmians, and the space of all  Boehmiansis denoted by 𝔅.The sum of two Boehmians and multiplication by a scalar can be defined in a natural way(3.2)[fnϕn]+[gnψn]=[((fn⋅ψn)+(gn⋅ϕn))ϕn⋅ψn],α[fnϕn]=[αfnϕn],α∈C.
The operation · and the differentiation are defined by(3.3)[fnϕn]⋅[gnψn]=[(fn⋅gn)(ϕn⋅ψn)],Dα[fnϕn]=[Dαfnϕn].
The relationship between the notion of convergence and the product · are given by the following:(i)
iffn→f as n→∞ in 𝔛 and, ϕ∈ℑ is any fixed element, then fn·ϕ→f·ϕ in 𝔛(asn→∞),(ii)
iffn→f as n→∞ in 𝔛 and (δn)∈Δ, then fn·δn→f in 𝔛(asn→∞).The operation· can be extended to 𝔅×ℑ by(3.4)If[fnδn]∈Bandϕ∈I,then[fnδn]⋅ϕ=[fn⋅ϕδn].
In 𝔅, one can define two types of convergence as follows:(i)
(δ-convergence) a sequence(βn)in𝔅is said to beδ-convergent toβin𝔅,denoted byβn→δβ,if there exists a delta sequence(δn)such that(βn·δn),(β·δn)∈𝔛,forallk,n∈N,   and(βn·δk)→(β·δk)  asn→∞,  in𝔛,   for everyk∈N,(ii)
(Δ-convergence) a sequence(βn)in𝔅is said to beΔ-convergent toβin𝔅,denoted byβn→Δβ,if there exists a(δn)∈Δsuch that(βn-β)·δn∈𝔛,foralln∈N,and(βn-β)·δn→0asn→∞in𝔛.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 𝔅L1 the space of Lebesgue integrable Boehmians [17] with the set Δ of all delta sequence (δn) from 𝔇 (the test function space of compact support) such that(1)
∫Rδn=1 for all n∈N,(2)
∫R|δn|<M for certain positive number M and n∈N,(3)
∫|t|>ε|δn(t)|dt→0 as n→∞ for every ε>0.Then,𝔅L1 is a convolution algebra with the pointwise operations(i)
λ[fn/δn]=[λfn/δn],(ii)
[fn/δn]+[gn/ϕn]=[(fn*ϕn+gn*δn)/(δn*ϕn)],(iii)
and the convolution(3.5)[fnδn]*[gnϕn]=[fn*gnδn*ϕn].Lemma 3.1.
Let[fn/δn]∈𝔅L1, then the sequence
(3.6)Fd(fn(t);ξ)=12πiγ1∫Rfn(t)exp⁡(i2γ1(α1t2-2tξ+α2ξ2))dt
converges uniformly on each compact set  K in  R.Proof.
Letf̃n=𝔉df. For each compact set K,δ̃n(δ̃n=𝔉dδn) converges uniformly to the function exp(-(iα2/2γ1)ξ2).  Hence, by Corollary 2.7,
(3.7)Fd(fn(t);ξ)=f̃nδ̃kδ̃k=e(iα2/2γ1)ξ22πiγ1Fd(fn*δk)δ̃k.
Using the choice fn/δn that is quotient of sequences and upon employing Corollary 2.7, we have
(3.8)Fd(fn(t);ξ)=e(iα2/2γ1)ξ22πiγ1Fd(fk*δn)δ̃k=f̃kδ̃kδ̃n=f̃kδ̃k2πiγ1e-(iα2/2γ1)ξ2.
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:[fn/δn]  in𝔅L1 as (3.9)R[fnδn]=lim⁡n→∞⁡f̃n,where the limit ranges over compact subsets ofR. Now, let [Xn/δn]=[Yn/γn] in 𝔅L1, then(3.10)Xn*γm=Ym*δn,foreverym,n∈N.
Hence, employing the Fresnel transform to both sides of above equation implies(3.11)Fd(Xn*γm)=Fd(Ym*δn)=Fd(Yn*δm).
Thus, using Theorem 2.6 and the fact that(3.12)δ̃nandδ̃m⟶2πiγ1e-(iα2/2γ1)ξ2,
on compact subsets of R, we get(3.13)lim⁡n→∞FdXn=lim⁡n→∞FdYn.
Hence,(3.14)R[Xnδn]=R[Ynγn].
The definition is therefore well defined.Theorem 3.2.
LetB1 and  B2 be in  𝔅L1 and  α∈ℂ,  then(i)
ℛ(αB1)=αℛB1,(ii)
ℛ(B1+B2)=ℛB1+ℛB2,(iii)
ℛ(B1*δn)=2πiγ1e-(iα2/2γ1)ξ2ℛB1=ℛ(δn*B1),(iv)
ifℛB1=0, then  B1=0,(v)
ifBn→ΔB as  n→∞ in  ℬL1, then  ℛBn→ΔℛB as  n→∞ in  𝔅L1 on compact subsets.Proof.
The proof of (i), (ii), and (iv) follows from the corresponding properties of the distributional Fresnel transform. Since eachf∈E´ has a representative
(3.15)f⟶[f*ϕnϕn],
in the space 𝔅L1, 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 transformℛ is continuous with respect to the  δ-convergence.Proof.
LetBn→δB in BL1 as n→∞, then we show that ℛBn→δℛB as n→∞. Using [17, Theorem  2.6], we find [fn,k/δk]=Bn and [fk/δk]=B such that fn,k→fk as n→∞,k∈N. Applying the Fresnel transform for both sides implies f̃n,k→f̃k in the space of continuous functions. Therefore, considering limits, we get
(3.16)R[fn,kδk]⟶R[fkδk].
This completes the proof of the theorem.Theorem 3.4.
The diffraction Fresnel transformℛ  is continuous with respect to the  Δ-convergence.Proof.
LetBn→ΔB as n→∞ in n→∞, then there is fn∈L1 and δn∈Δ such that
(3.17)(Bn-B)*δn=[fn*δnδk],fn→0asn→∞.
Thus
(3.18)R((Bn-B)*δn)=R[fn*δnδk]⟶Fd(fn*δn)asn→∞⟶2πiγ1e-(iα2/2γ1)ξ2Fdfnasn⟶∞byCorollary2.7⟶0bythelinearityofFdfn.
Therefore, ℛ(Bn-B)→0 as n→∞. Thus, ℛBn→ΔℛB as n→∞. This completes the proof.Lemma 3.5.
Let[fn/ϕn]∈ℬL1 and  ϕ∈𝔇(R), then
(3.19)R([fnϕn]*ϕ)=2πiγ1ei(2α1tτ-α2ξ2)/2γ1R[fnϕn]*Fdϕ.Proof.
Let[fn/ϕn]∈𝔅L1, then using (3.9), we have
(3.20)R([fnϕn]*ϕ)=R[fn*ϕϕn]=lim⁡n→∞Fd(fn*ϕ),
on compact subsets of R. By applying Theorem 2.6, it yields
(3.21)R([fnϕn]*ϕ)=2πiγ1ei(2α1tτ-α2ξ2)/2γ1lim⁡n→∞Fd(f(t);ξ)Fd(ϕ(τ);ξ).
Hence, ℛ([fn/ϕn]*ϕ)=2πiγ1ei(2α1tτ-α2ξ2)/2γ1ℛ[fn/ϕn]𝔉d(ϕ(τ);ξ). This completes the proof of the lemma.



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