Research Article On Diffraction Fresnel Transforms for Boehmians

Transcription

1 Abstract and Applied Analysis Volume 20, Article ID 72746, pages doi:0.55/20/72746 esearch Article On Diffraction Fresnel Transforms for Boehmians S. K. Q. Al-Omari and A. Kılıçman 2 Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 34, Jordan 2 Department of Mathematics and Institute of Mathematical esearch, Universiti Putra Malaysia UPM), Serdang, Selangor, Malaysia Correspondence should be addressed to A. Kılıçman, akilicman@putra.upm.edu.my eceived 2 September 20; evised 27 October 20; Accepted November 20 Academic Editor: Natig Atakishiyev Copyright q 20 S. K. Q. Al-Omari and A. Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.. 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 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. ecently, much attention has been paid to the diffraction Fresnel transform f ξ K α,γ,γ 2,α 2 ; ξ, t ) f t dt,. where K α,γ,γ 2,α 2 ; ξ, t ) i exp α t 2 2ξt α 2 ξ 2)) 2πiγ.2

2 2 Abstract and Applied Analysis is the transform kernel with the real parameters and α,γ,andγ 2 satisfy the following relation: α α 2 γ γ 2.3 holds; see 9. Many familiar transforms can be considered as special cases of the generalized Fresnel transform. For example, if the parameters α,γ,γ 2 and α 2 satisfy the matrix ) α γ γ 2 α 2 ) cos θ sin θ sin θ cos θ.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 α α 2, 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 / t. When t is one dimensional, every function φ t in S satisfies the infinite set of inequalities t m φ k t C m,k, where t, 2. where m and k run through all nonnegative integers. The above expression can be interpreted as lim t tm φ k t 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, 0,. Theorem 2.. If φ t is in S, then its diffraction Fresnel transform φ ) ξ 2πiγ i α t 2 2tξ α 2 ξ 2) ) φ t exp dt 2.3 exists and further also in S.

3 Abstract 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, k- times, yields a sum of polynomials, p k t ξ, say of combinations of t and ξ. Thatis, d k dt F ) i k d φ ξ p α t 2 2tξ α 2 ξ 2) ) k t ξ φ t exp pk t ξ φ t, 2.4 which is also in S,sinceφ in S and S is a linear space. Hence, dt F ) k d φ ξ ξm dk ξ m p 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,then f x g x dx f x g x dx, 2.6 where f and g are the corresponding diffraction Fresnel transforms of f and g, respectively. Proof. The diffraction Fresnel transforms f ξ and g ξ are indeed bounded and continuous for all ξ. This ensure the convergence of the integrals in 2.6. Moreover, f x g x dx dx f x g y ) i α y 2 2xy α 2 x 2) ) exp dy, α α 2 γ γ 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 f x g y ) i α2 x 2 2xy α y 2) ) exp dy dx f y ) g y ) dy, 2.8 where α 2 α γ γ 2. This completes the proof of the theorem. Parseval s relation can be interpreted as Fd f, φ f, φ. 2.9

4 4 Abstract and Applied Analysis Therefore, from the above relation, we state the diffraction Fresnel transform of a distribution f of slow growth f Ś as Fd f, φ f, φ, φ S, 2.0 and it is well defined by Theorem 2.. Theorem 2.3. If f is a distribution of slow growth, then its diffraction Fresnel transform f is also a distribution of slow growth. Proof. Linearity of f is obvious. To show continuity of f,let n n 0inSas n. Hence, 0, in S, then also Fd f, f, Fd 0 as n. 2. Hence 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 i α t f ξ f t, 2 2tξ α 2 ξ 2) ) exp πiγ Proof. Let φ S be arbitrary. From 2.0,weread Fd f ξ,φ ξ f t, φ t i α ξ f t, 2 2tξ α 2 t 2) ) φ ξ exp dξ 2πiγ i α2 t 2 2tξ α ξ 2) ) f t, exp φ ξ dξ 2πiγ i α2 t f t, 2 2tξ α ξ 2) ) exp,φ ξ. 2πiγ 2.3 But since f t, exp i α 2 t 2 2tξ α ξ 2 / is an infinitely smooth function, we get i α2 t f ξ f t, 2 2tξ α ξ 2) ) exp πiγ This completes the proof of the theorem. Now, for distributions f and g É, we define the convolution product as f 9 ) t,φ t f t, g τ,φ t τ, 2.5

5 Abstract and Applied Analysis 5 for every φ E. This definition makes sense, since g τ,φ t τ belongs to D, and hence amemberofe. With this definition, we are allowed to write the following theorem. Theorem 2.5. For every f É, the function ψ t f τ,φ t τ is infinitely smooth and satisfies the relation Dt k ψ t f τ,dt, k φ t τ 2.6 for all k N. Proof see page 26 in 2). 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 ξ f t ; ξ, g ξ g τ ; ξ their respective diffraction Fresnel transforms, then ) ) i 2α tτ α 2 ξ f g t ; ξ 2πiγ 2) ) ) ) exp f t ; ξ Fd g τ ; ξ. 2.7 Proof. Let f, g É, then by using 2.2, weget ) ) f ) i α t 2 2tξ α 2 ξ 2) ) f g t ; ξ g t, exp 2πiγ i α t τ i.e. f t, 2 t τ ξ α 2 ξ 2) g τ, exp 2πiγ i α t f t, 2 α τ 2 α tτ 2tξ 2τξ α 2 ξ 2) ) g τ, exp. 2πiγ 2.8 Properties of distributions together with simple calculations on the exponent yield ) ) i 2α tτ α 2 ξ f g t ; ξ 2πiγ 2) ) ) ) exp f t ; ξ Fd g τ ; ξ. 2.9 This completes the proof of the theorem. Corollary 2.7. Let f, g É,then 2 f δn t ; ξ ) δn g t ; ξ ) 2πiγ exp α ) 2ξ 2 ) f ξ, 2πiγ exp α ξ 2 ) ) g ξ, where f ξ f t ; ξ, g ξ g τ ; ξ.

6 6 Abstract and Applied Analysis The following is a theorem which can be directly established from 2.2 and the fact that D k f g ) D k f g f D k g. 2.2 Theorem 2.8. Let f and g be distributions of compact support and f ξ f t ; ξ, g ξ g τ ; ξ their respective diffraction Fresnel transforms, then Dt k ) ) i 2α tτ α 2 ξ 2) ) ) f g t ; ξ 2πiγ exp f k t ; ξ g ξ, 2 Dt k ) ) i 2α tτ α 2 ξ 2) ) ) f g t ; ξ 2πiγ exp f ξ g k τ ; ξ 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,weassign 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, 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, 2,..., then f g, b if, δ n Δ, then ψ n Δ. Elements of Δ will be called delta sequences. Consider the class U of pair of sequences defined by U { ), φn )) : ) X N, ) Δ }, 3. for each n N. An element f n, U is called a quotient of sequences, denoted by f n /,or f n / if f i φ j f j φ i, for all i, j N. Similarly, two quotients of sequences f n / and g n /ψ n are said to be equivalent, f n / g n /ψ n,iff i ψ j g j φ i, for all i, j N. Therelation is an equivalent relation on U, and hence splits U into equivalence classes. The equivalence class containing f n / is denoted by f n /. These equivalence classes are called Boehmians, andthespace of all Boehmians is denoted by B. The sum of two Boehmians and multiplication by a scalar can be defined in a natural way gn α ψ n ) )) ψ n gn, ψ n α, α C. 3.2

7 Abstract and Applied Analysis 7 The operation and the differentiation are defined by gn ψ n D α ) g n ), φn ψ n D α f 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 as n, ii if f n f as n in X and δ n Δ, then f n δ n f in X as n. The operation can be extended to B I by If δ n B and φ I, then δ n φ φ δ 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 δ β n β, if there exists a delta sequence δn such that β n δ n, β δ n X, for all k, 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 B is said to be Δ-convergent to β in B, denoted by Δ β n β, if there exists a δn Δ such that β n β δ n X, for all n N, and β n β δ n 0 as n in X. For further analysis we refer, for example, to 0, 3 9. NowweletL be the space of Lebesgue integrable functions on and B L the space of Lebesgue integrable Boehmians 7 with the set Δ of all delta sequence δ n from D the test function space of compact support such that δ n for all n N, 2 δ n <Mfor certain positive number M and n N, 3 t >ε δ n t dt 0asn for every ε>0. Then, B L 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 / f n g n δ n / δ n, iii and the convolution δ n gn g n. 3.5 δ n

8 8 Abstract and Applied Analysis Lemma 3.. Let f n /δ n B L, then the sequence t ; ξ ) 2πiγ i f n t exp α t 2 2tξ α 2 ξ 2)) dt 3.6 converges uniformly on each compact set K in. Proof. Let f n f. For each compact set K, δn δ n δ n converges uniformly to the function exp iα 2 / ξ 2. Hence, by Corollary 2.7, t ; ξ ) ) f δk F n e iα2/2γ ξ2 d δ k. 3.7 δ k 2πiγ δ k Using the choice f n /δ n that is quotient of sequences and upon employing Corollary 2.7, we have t ; ξ ) e iα 2/ ξ 2 2πiγ fk δ n ) δ k f k δ k δn f k δ k 2πiγ e iα 2/ ξ 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 L as δ n lim n, 3.9 where the limit ranges over compact subsets of. Now,let X n /δ n Y n /γ n in B L, then X n γ m Y m δ n, for every m, n N. 3.0 Hence, employing the Fresnel transform to both sides of above equation implies Xn γ m ) Fd Y m δ n Y n δ m. 3.

9 Abstract and Applied Analysis 9 Thus, using Theorem 2.6 and the fact that δ n and δ m 2πiγ e iα 2/ ξ 2, 3.2 on compact subsets of,weget lim n X n lim n Y n. 3.3 Hence, Xn δ n Yn γ n. 3.4 The definition is therefore well defined. Theorem 3.2. Let B and B 2 be in B L and α C, then i αb αb, ii B B 2 B B 2, iii B δ n 2πiγ e iα 2/ ξ 2 B δ n B, iv if B 0, thenb 0, v if B n Δ B as n in BL,thenB n Δ B as n in BL 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 φn f, 3.5 in the space B L,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 7, Theorem 2. This completes the proof of the theorem. Theorem 3.3. The Fresnel transform is continuous with respect to the δ-convergence. δ δ Proof. Let B n B in BL as n, then we show that B n Bas n.using 7, Theorem 2.6, wefind f n,k /δ k B n and f k /δ k Bsuch 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,k δ k fk δ k. 3.6 This completes the proof of the theorem.

10 0 Abstract and Applied Analysis Theorem 3.4. The diffraction Fresnel transform is continuous with respect to the Δ-convergence. Proof. Let B n Δ B as n in n, then there is L and δ n Δ such that δ n B n B δ n, f n 0asn. 3.7 δ k Thus δ n B n B δ n δ k ) δ n as n 2πiγ e iα 2/ ξ 2 f n as n by Corollary by the linearity of f n. Therefore, B n B 0asn.Thus,B n proof. Δ B as n. This completes the Lemma 3.5. Let f n / B L and φ D,then ) φ 2πiγ e i 2α tτ α 2 ξ 2 / φ. 3.9 Proof. Let f n / B L, then using 3.9, we have ) φ φ lim n φ ), 3.20 on compact subsets of. By applying Theorem 2.6, it yields ) φ 2πiγ e i 2α tτ α 2 ξ 2 / ) ) lim f t ; ξ Fd φ τ ; ξ. n 3.2 Hence, f n / φ 2πiγ e i 2α tτ α 2 ξ 2 / f n / 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 esearch University Grant Scheme no U.

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