Approximation by quasi-interpolation operators and Smolyak's algorithm

2020/12/15 by Kolomoitsev, Yurii
#41A17 #41A25 #41A58 #41A63 #42A10 #42A15 #42B25 #42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2012.08273

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 B_p,θ^s(\mathbbT^d) and the Triebel--Lizorkin spaces F_p,θ^s(\mathbbT^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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