Approximation by quasi-interpolation operators and Smolyak's algorithm

We study approximation of multivariate periodic functions from Besov and Triebel–Lizorkin spaces dominating mixed smoothness by the Smolyak algorithm constructed using a special class quasi-interpolation operators Kantorovich-type. These are defined similar to classical sampling replacing samples with average values function on small intervals (or more generally sampled convolution given an appropriate kernel). In this paper, we estimate rate convergence corresponding in Lq-norm for Bp,θs(Td) Fp,θs(Td) all s>0 admissible 1≤p,θ≤∞ as well provide analogues Littlewood–Paley-type characterizations these terms families operators.