Approximation by periodic multivariate quasi-projection operators

Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions ${\phi }_j$ and a sequence of distributions/functions ${\tilde{\phi }}_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for ${\phi }_j$ and the compatibility conditions for ${\phi }_j$ and ${\tilde{\phi }}_j$. 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.