Approximation by multivariate quasi-projection operators

Quasi-projection operators with a matrix dilation $M$ are
$$Q_j (f, \phi, \tilde{\phi}) = \sum_{k\in\mathbb{Z}^d}\langle f, \tilde{\phi}_{jk}\rangle \phi_{jk},$$
where $\phi$ is a function, and $\tilde{\phi}$ is a function or a tempered distribution,
$$\psi_{jk}(x) := m^{j/2}\psi(M^jx + k), \qquad j \in \mathbb{Z}, k \in \mathbb{Z}^d,$$
$M$ is a $d \times d$ matrix whose eigenvalues are bigger than $1$ (in absolute value), $m = |\mathrm{det} M|$.
We consider different classes of such operators and study their approximation properties. Error estimates in $L_p$ -norm, $2 \le p \le \infty$, are provided for a large class of functions $\phi$ (including both band-limited and compactly supported functions) and for $\tilde{\phi} \in \mathcal{S}'_N$ , where $\mathcal{S}'_N$ is the set of tempered distribution whose Fourier transform $\hat{\tilde{\phi}}$ is a function on $\mathbb{R}^d$ such that $|\hat{\tilde{\phi}}(\xi)| \le C_{{\tilde{\phi}}}|\xi|^N$ for almost all $\xi \notin \mathbb{T}^d , N = N({\tilde{\phi}}) \ge 0$, and $|\hat{\tilde{\phi}}(\xi)| \le C'_{\phi}$ for almost all $\xi \in \mathbb{T}^d$. The estimates are given in terms of the Fourier transform of $f$. In particular, a finite linear combination of the Dirac delta-function and its derivatives is in $\mathcal{S}'_N$. If $\tilde{\phi}$ is the Dirac delta-function and $\phi = \mathrm{sinc}$, then $Q_j (f, \phi, \tilde{\phi})$ is the classical sampling operator.
Another class of quasi-projection operators we study includes classical Kantorovich – Kotelnikov operators, where $\tilde{\phi}$ is the characteristic function of $[0, 1]$. In this case $\langle f, \tilde{\phi}_{jk}\rangle$ is the averages value of $f$ near the node $M^{-j}k$ (instead of the exact value $f(M^{-j}k)$ in the sampling expansion), which allows to deal with discontinues signals and reduce the so-called time-jitter errors. Error estimates in $L_p$-norm, $1 \le p \le \infty$, for this class are given in terms of classical moduli of smoothness. Such estimates are aimed at the recovery of signals $f$, but they are not applicable to non-decaying signals and for signals whose decay is not enough to be in $L_p$, which are of interest to engineers. However such signals may belong to a weighted $L_p$ space. Error estimates in the weighted $L_p$ spaces are also obtained for the Kantorovich – Kotelnikov-type and sampling operators.