This note focuses on state observer design for a general class of nonlinear discrete-time systems. The main contribution lies in the use of the differential mean value theorem (DMVT) to transform the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general condition on the non linear functions. An extension to H_∞ filtering design is obtained for systems with linear and nonlinear outputs. LMI conditions are presented to ensure asymptotic convergence. Then performances and accuracy of the results are illustrated through simulation examples.