We proved direct and inverse theorems on B-spline quasi-interpolationsampling representation with a Littlewood-Paley-type norm equivalence inSobolev spaces $W^r_p$ of mixed smoothness $r$, established estimates of theapproximation error of recovery in $L_q$-norm of functions from the unit ball$U^r_p$ in the spaces $W^r_p$ by linear sampling algorithms based on thisrepresentation, the asymptotic optimality of these sampling algorithms in termsof Smolyak sampling width $r^s_n(U^r_p, L_q)$ and sampling width $r_n(U^r_p,L_q)$.
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机译：我们在混合光滑度$ r $的Sobolev空间$ W ^ r_p $的Littlewood-Paley型范数等效的B样条拟插值采样表示中证明了正定和逆定理，建立了$ L_q $ -norm中恢复近似误差的估计基于此表示的线性采样算法对空间$ W ^ r_p $中单位球$ U ^ r_p $的函数的求和，这些采样算法的Smolyak采样宽度$ r ^ s_n（U ^ r_p，L_q）的渐近最优性$和采样宽度$ r_n（U ^ r_p，L_q）$。
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