We prove maximum and comparison principles for the discrete fractional derivatives in the integers. Regularity results when the space is a mesh of length h, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough (Hölder continuous), these approximation procedures give a measure of the order of approximation. These results also allow us to prove the coincidence, for Hölder continuous functions, of the Marchaud and Grünwald–Letnikov derivatives in every point and the speed of convergence to the Grünwald–Letnikov derivative. The discrete fractional derivative will be also described as a Neumann–Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces lp(Z)
