Approximation by quasi-interpolation operators and Smolyak's algorithm

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 Lq-norm for functions from the Besov spaces Bsp,theta(Td) and the Triebel-Lizorkin spaces Fsp,theta(Td) for all s > 0 and admissible 1 < p, theta < infinity as well as provide analogues of the Littlewood- Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators. (c) 2021 Elsevier Inc. All rights reserved.