  In this paper, we propose and analyse a reconstruction technique which
enables one to design high-order conservative semi-Lagrangian schemes for
kinetic equations. The proposed reconstruction can be obtained by taking the
sliding average of a given polynomial reconstruction of the numerical solution.
A compact representation of the high order conservative reconstruction in one
and two space dimension is provided, and its mathematical properties are
analyzed. To demonstrate the performance of proposed technique, we consider
implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin
model and the Broadwell model, and then solve related shock problems which
arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations
will be presented in the second part of the paper.
