The Fresnel transform has been used in several applications of digital holography to recover wave fields from digital holograms. Three-dimensional (3D) reconstruction, 3D recognition and particle tracking velocimetry can be found among these applications. Depending on the application, the recovered wave fields should satisfy certain requirements. Sampling rate control (availability to choose the distance between samples in the recovered wave field) is a requirement for several digital holography applications; furthermore, recovery region control (availability to choose the size and position of the recovered wave fields) can be useful since in most applications only a small region of the wave fields is required. Nevertheless, none of the actual formulations of the Fresnel transform can be used to control the sampling rate and the recovery region in the same formulation. There are a few proposals that completely control the sampling rate of the wave fields to be recovered, but they use the computation of at least a couple of two dimensional discrete Fourier transforms with dependency between them. This dependency restricts the minimum execution time that can be achieved. In this research, it is proved that implementations of the Fresnel transform with a single two-dimensional discrete Fourier transform can be used to control the sampling rate and the recovery region of the wave fields and, at the same time, to reduce the required execution time. In a proposed software alternative, the use of a single two-dimensional discrete Fourier transform can achieve shorter execution times for most of the practical applications than the current alternatives if a small flexibility is permitted in the required sampling rate. Furthermore, a parallel hardware architecture, where the flexibility is not required, is proposed. The hardware architecture can achieve shorter execution times than any existing alternative to compute the Fresnel transform. The new formulation of the Fresnel transform can require computing only a few coefficients of the two-dimensional discrete Fourier transform applied to an input array padded with zeros. In order to reduce the execution time required by the new formulation of the Fresnel transform, an input and/or output pruning method for composite length discrete Fourier transforms was proposed. The pruning method avoids computing the multiplications per zero and non required Fourier coefficients.