Multivariate approximation by translates of the Korobov function on Smolyak grids

TL;DR

This paper investigates multivariate function approximation on the torus using translates of the Korobov function, employing sparse Smolyak grids to achieve efficient error bounds, especially in the Hilbert space case.

Contribution

It provides new upper bounds for approximation errors using translates of the Korobov function on Smolyak grids, including constructions for the case of Hilbert spaces.

Findings

Established upper bounds for worst-case approximation errors.

Constructed approximation methods based on sparse Smolyak grids.

Provided lower bounds for optimal approximation with Korobov functions.

Abstract

For a set $\mathbb{W} \subset L_p(\bT^d)$, $1 < p < \infty$, of multivariate periodic functions on the torus $\bT^d$ and a given function $\varphi \in L_p(\bT^d)$, we study the approximation in the $L_p(\bT^d)$-norm of functions $f \in \mathbb{W}$ by arbitrary linear combinations of $n$ translates of $\varphi$. For $\mathbb{W} = U^r_p(\bT^d)$ and $\varphi = \kappa_{r,d}$, we prove upper bounds of the worst case error of this approximation where $U^r_p(\bT^d)$ is the unit ball in the Korobov space $K^r_p(\bT^d)$ and $\kappa_{r,d}$ is the associated Korobov function. To obtain the upper bounds, we construct approximation methods based on sparse Smolyak grids. The case $p=2, \ r > 1/2$, is especially important since $K^r_2(\bT^d)$ is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by $\kappa_{r,d}$. We also provide lower bounds of the optimal approximation on the best choice of $\varphi$.