A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms avoiding the notion of fluxes and artificial penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the H-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem.