Approximation by quasi-interpolation operators and Smolyak's algorithm

TL;DR

This paper investigates the approximation of multivariate periodic functions in Besov and Triebel--Lizorkin spaces using Smolyak's algorithm with quasi-interpolation operators, providing convergence rates and characterizations.

Contribution

It introduces a new approach to approximation using Kantorovich-type quasi-interpolation operators within Smolyak's algorithm, extending convergence analysis to broad function spaces.

Findings

Derived convergence rates of the Smolyak algorithm in $L_q$-norm.

Established Littlewood--Paley-type characterizations using quasi-interpolation.

Extended approximation results to all $s>0$ and admissible $p, heta$.

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 $\mathbf{B}_{p,\theta}^s(\mathbb{T}^d)$ and the Triebel--Lizorkin spaces $\mathbf{F}_{p,\theta}^s(\mathbb{T}^d)$ for all $s>0$ and admissible $1\le p,\theta\le \infty$ as well as provide analogues of the Littlewood--Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.