Approximation by quasi-interpolation operators and Smolyak's algorithm
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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 𝐁_p,θ^s(𝕋^d) and the Triebel–Lizorkin spaces 𝐅_p,θ^s(𝕋^d) 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.
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