On Diffraction Fresnel Transforms for Boehmians

and Applied Analysis 3 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, ktimes, yields a sum of polynomials, pk t ξ , say of combinations of t and ξ. That is, ∣ ∣ ∣ ∣ ∣ d dtk Fd ( φ ) ξ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ pk t ξ φ t exp ( i ( α1t 2 − 2tξ α2ξ ) 2γ1 )∣ ∣ ∣ ∣ ≤ ∣pk t ξ φ t ∣ ∣, 2.4 which is also in S, since φ in S and S is a linear space. Hence, ∣ ∣ ∣ ∣ ∣ ξ d dtk Fd ( φ ) ξ ∣ ∣ ∣ ∣ ∣ ≤ ∫ R ∣ ∣ξpk 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 Cm,k, for every pair of nonnegative integersm 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


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 where Abstract and Applied Analysis is the transform kernel with the real parameters and α 1 , γ 1 , and γ 2 satisfy the following relation: Many familiar transforms can be considered as special cases of the generalized Fresnel transform.For example, if the parameters α 1 , γ 1 , γ 2 and α 2 satisfy the matrix then the generalized Fresnel transform becomes a fractional Fourier transform.
In particular, when θ π/2, one obtains the standard Fourier transform.Further, if α 1 α 2 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.

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 inequalities where m and k run through all nonnegative integers.The above expression can be interpreted as 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 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, ktimes, yields a sum of polynomials, p k t ξ , say of combinations of t and ξ.That is, which is also in S, since φ in S and S is a linear space.Hence, 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 where F d f and F d g are the corresponding diffraction Fresnel transforms of f and g, respectively.
Proof.The diffraction Fresnel transforms F d f ξ and F d g ξ are indeed bounded and continuous for all ξ.This ensure the convergence of the integrals in 2.6 .Moreover,

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 where α 2 α 1 − γ 1 γ 2 1.This completes the proof of the theorem.
Parseval's relation can be interpreted as Therefore, from the above relation, we state the diffraction Fresnel transform of a distribution f of slow growth f ∈ Ś as and it is well defined by Theorem 2.1.
Hence F d f ∈ Ś.This completes the proof of the theorem.
Theorem 2.4.Let f be a distribution of compact support f ∈ É .Then, we define the Fresnel transform of f as Proof.Let φ ∈ S R be arbitrary.From 2.10 , we read

2.13
But since f t , exp i α 2 t 2 −2tξ α 1 ξ 2 /2γ 1 is an infinitely smooth function, we get This completes the proof of the theorem.Now, for distributions f and g ∈ É R , we define the convolution product as 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 ∈ É R , the function ψ t f τ , φ t τ is infinitely smooth and satisfies the relation 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 ξ their respective diffraction Fresnel transforms, then Proof.Let f, g ∈ É R , then by using 2.12 , we get

2.18
Properties of distributions together with simple calculations on the exponent yield This completes the proof of the theorem.

Abstract and Applied Analysis
The following is a theorem which can be directly established from 2.12 and the fact that [11]

2.21
Theorem 2.8.Let f and g be distributions of compact support and 2.22

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: Elements of Δ will be called delta sequences.Consider the class U of pair of sequences defined by Similarly, two quotients of sequences f n /φ n and g n /ψ n are said to be equivalent, 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 way

3.2
Abstract and Applied Analysis 7 The operation • and the differentiation are defined by

3.3
The relationship between the notion of convergence and the product • are given by the following: The operation • can be extended to 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 ii Δ-convergence a sequence β n in B is said to be Δ-convergent to β in B, denoted by for all n ∈ N, and For further analysis we refer, for example, to 10, 13-19 .Now we let L 1 be the space of Lebesgue integrable functions on R and B L 1 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 Then, B L 1 is a convolution algebra with the pointwise operations iii and the convolution converges uniformly on each compact set K in R.
Proof.Let f n F d f.For each compact set K, δ n δ n F d δ n converges uniformly to the function exp − iα 2 /2γ 1 ξ 2 .Hence, by Corollary 2.7, Using the choice f n /δ n that is quotient of sequences and upon employing Corollary 2.7, we have 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 , 3.9 where the limit ranges over compact subsets of R. Now, let for every m, n ∈ N.

3.10
Hence, employing the Fresnel transform to both sides of above equation implies

3.11
Abstract and Applied Analysis 9 Thus, using Theorem 2.6 and the fact that δ n and δ m −→ 2πiγ 1 e − iα 2 /2γ 1 ξ 2 , 3.12 on compact subsets of R, we get The definition is therefore well defined.
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 in the space B L 1 , 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.
Applying the Fresnel transform for both sides implies f n,k → f k in the space of continuous functions.Therefore, considering limits, we get This completes the proof of the theorem.