  Multivariate quasi-projection operators $Q_j(f,\varphi,
\widetilde{\varphi})$, associated with a function $\varphi$ and a
distribution/function $\widetilde{\varphi}$, are considered. The function
$\varphi$ is supposed to satisfy the Strang-Fix conditions and a compatibility
condition with $\widetilde{\varphi}$. Using technique based on the Fourier
multipliers, we studied approximation properties of such operators for
functions $f$ from anisotropic Besov spaces and $L_p$ spaces with $1\le p\le
\infty$. In particular, upper and lower estimates of the $L_p$-error of
approximation in terms of moduli of smoothness and best approximations are
obtained.
