Abstract: The Fresnel transform is applied on an ideal wavelet basis for (ℝ), the -splines [1], which results into the -splines.In digital holography the image recorded on a has to be reconstructed. Though the underlying idea of Fresnelets are given in [2], a detailed mathematical derivation is provided here. As a special instance of Fresnelets, the Franklin Fresnelets are considered. To meet the requirements of retrieving the image,the Fresnelets are more effective than the wavelets derived from the -spline scaling functions.