Sufficient conditions for the application of the Fresnel approximation of the two-dimensional Kirchhoff diffraction formula are derived for spatially band-limited input signals diffracted by a slit. The method used is an extension of the one given by Papoulis, who derives conditions only for input signals that are sufficiently smooth. For band-limited input signals, the Fresnel region can have an upper limit for the axial extent, depending on the maximum frequency in the input signal.
P. Matthijsse and P. J. G. Hammer, "Fresnel region for spatially band-limited signals," J. Opt. Soc. Am. 65, 188-190 (1975)