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.
翻译:我们从Besov 和 Triebel-Lizorkin 中研究由Smolyak 算法, 利用Kantorovich- 类型的特殊类准内插操作员, 构建的Smolyak 算法, 支配混合平滑的多变量周期空间的近似值。 这些操作员的定义类似于古典采样操作员。 这些操作员的定义是, 将样本替换为以小间隔( 或更一般地以特定函数与适当内核相融合的样本值 ) 函数的平均值 。 在本文中, 我们估计了来自 Besov 空间 $\ q$\ q$ 的对应的 Smolyak 算法的趋近率, 以 $\ q $ 为单位, 和 可受理的 $ 1\ le p,\ ta\ le\ le\ mintlefty$, 以及提供 littlewood- place- typecal- typeal- sistrical- typecal- typecal- sistrations) 的模拟操作员家族的类比木- 的模拟操作员- 的类比木- 的类比木- 的类比木- 定的模拟操作员- 。