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 )∣ ∣...
