The aim of this work is an advancement of digital holographic interferometry by using a digital micromirror device (DMD) for real-time display of interferometric fringes. The theory describing the formation of the diffracted image produced by a coherently illuminated DMD is presented. Digital holograms of simple structures are calculated assuming plane and spherical reference waves, respectively. These synthetic holograms are used to control the micromirrors of the DMD, which now acts as a reflective hologram. Experimental holographic reconstructions of synthetic holograms prove the feasibility of DMDs in principle for this aim. Further experimental results show the reconstructed interferogram of fields recorded by digital holography and superposed numerically. This opens a new range of applications in digital holographic interferometry.