# Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

## TL;DR

This paper investigates the computational complexity of multivariate L2-approximation in hybrid spaces combining Walsh and Korobov functions, analyzing how the difficulty scales with dimension and information class.

## Contribution

It provides conditions for various levels of tractability of L2-approximation in hybrid function spaces, considering all linear functionals and standard evaluations.

## Key findings

- Conditions for weak tractability established
- Criteria for polynomial and strong polynomial tractability derived
- Analysis of the impact of function space structure on approximation complexity

## Abstract

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.