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 $φ_j$ and a sequence of distributions/functions $\widetildeφ_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for $φ_j$ and the compatibility conditions for $φ_j$ and $\widetildeφ_j$. 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.

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