  Approximation properties of periodic quasi-projection operators with matrix
dilations are studied. Such operators are generated by a sequence of functions
$\varphi_j$ and a sequence of distributions/functions $\widetilde{\varphi}_j$.
Error estimates for sampling-type quasi-projection operators are obtained under
the periodic Strang-Fix conditions for $\varphi_j$ and the compatibility
conditions for $\varphi_j$ and $\widetilde{\varphi}_j$. These estimates are
given in terms of the Fourier coefficients of approximated functions and
provide analogs of some known non-periodic results. Under some additional
assumptions error estimates are given in other terms in particular using the
best approximation. A number of examples are provided.
