B-spline quasi-interpolation sampling representation and sampling recovery in Sobolev spaces of mixed smoothness

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces $W_p^r$ of mixed smoothness $r$, established estimates of the approximation error of recovery in $L_q$-norm of functions from the unit ball $U_p^r$ in the spaces $W_p^r$ by linear sampling algorithms based on this representation, the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width $r_n^s(U_p^r,L_q)$ and sampling width $r_n(U_p^r,L_q)$.