An experimental and numerical study of new optical elements is outlined for the realization of differentiation as well as Hilbert and Fourier integral transforms of Fresnel images. The elements mentioned in the form of amplitude and phase gratings consist of two parts of different partition width-to-their disposition period ratios. A lensless mapping of two- dimensional Hilbert transform has been found experimentally by means of a grating grid consisting of two orthogonal gratings considered. The possibility is found of beam differentiation by means of a periodic grating permitting to eliminate the effect of the diffraction structure center shift in respect to the beam axis on the image formation. A substantial doubled image dependence on the input beam wavefront curvature has been revealed. The geometry of a two-dimensional spacial filter for optical beam switching is proposed.