  Differential and falsified sampling expansions $\sum_{k\in
\mathbb{Z}^d}c_k\phi(M^jx+k)$, where $M$ is a matrix dilation, are studied. In
the case of differential expansions, $c_k=Lf(M^{-j}\cdot)(-k)$, where $L$ is an
appropriate differential operator. For a large class of functions $\phi$, the
approximation order of differential expansions was recently studied. Some
smoothness of the Fourier transform of $\phi$ from this class is required. In
the present paper, we obtain similar results for a class of band-limited
functions $\phi$ with the discontinuous Fourier transform. In the case of
falsified expansions, $c_k$ is the mathematical expectation of random integral
average of a signal $f$ near the point $M^{-j}k$. To estimate the approximation
order of the falsified sampling expansions we compare them with the
differential expansions. Error estimations in $L_p$-norm are given in terms of
the Fourier transform of $f$.
