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