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 Lq-norm for functions from the Besov spaces Bp,θs(Td) and the Triebel–Lizorkin spaces Fp,θs(Td) for all s>0 and admissible 1≤p,θ≤∞ as well as provide analogues of the Littlewood–Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.