  We study approximation properties of general multivariate periodic
quasi-interpolation operators, which are generated by distributions/functions
$\widetilde{\varphi}_j$ and trigonometric polynomials $\varphi_j$. The class of
such operators includes classical interpolation polynomials
($\widetilde\varphi_j$ is the Dirac delta function), Kantorovich-type operators
($\widetilde\varphi_j$ is a characteristic function), scaling expansions
associated with wavelet constructions, and others. Under different
compatibility conditions on $\widetilde\varphi_j$ and $\varphi_j$, we obtain
upper and lower bound estimates for the $L_p$-error of approximation by
quasi-interpolation operators in terms of the best and best one-sided
approximation, classical and fractional moduli of smoothness, $K$-functionals,
and other terms.
