Approximation by quasi-interpolation operators and Smolyak's algorithm

Abstract

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.