  We study approximation of multivariate periodic functions from Besov and
Triebel--Lizorkin spaces of dominating mixed smoothness by the Smolyak
algorithm constructed using a special class of quasi-interpolation operators of
Kantorovich-type. These operators are defined similar to the classical sampling
operators by replacing samples with the average values of a function on small
intervals (or more generally with sampled values of a convolution of a given
function with an appropriate kernel). In this paper, we estimate the rate of
convergence of the corresponding Smolyak algorithm in the $L_q$-norm for
functions from the Besov spaces $\mathbf{B}_{p,\theta}^s(\mathbb{T}^d)$ and the
Triebel--Lizorkin spaces $\mathbf{F}_{p,\theta}^s(\mathbb{T}^d)$ for all $s>0$
and admissible $1\le p,\theta\le \infty$ as well as provide analogues of the
Littlewood--Paley-type characterizations of these spaces in terms of families
of quasi-interpolation operators.
