  The relation between the results of cosmological N-body simulations, and the
continuum theoretical models they simulate, is currently not understood in a
way which allows a quantification of N dependent effects. In this first of a
series of papers on this issue, we consider the quantification of such effects
in the initial conditions of such simulations. A general formalism developed in
[1] allows us to write down an exact expression for the power spectrum of the
point distributions generated by the standard algorithm for generating such
initial conditions. Expanded perturbatively in the amplitude of the input (i.e.
theoretical, continuum) power spectrum, we obtain at linear order the input
power spectrum, plus two terms which arise from discreteness and contribute at
large wavenumbers. For cosmological type power spectra, one obtains as
expected, the input spectrum for wavenumbers k smaller than that characteristic
of the discreteness. The comparison of real space correlation properties is
more subtle because the discreteness corrections are not as strongly localised
in real space. For cosmological type spectra the theoretical mass variance in
spheres and two point correlation function are well approximated {\it above} a
finite distance. For typical initial amplitudes this distance is a few times
the inter-particle distance, but it diverges as this amplitude (or,
equivalently, the initial red-shift of the cosmological simulation) goes to
zero, at fixed particle density. We discuss briefly the physical significance
of these discreteness terms in the initial conditions, in particular with
respect to the definition of the continuum limit of N-body simulations.
