A note on approximation in weighted Korobov spaces via multiple rank-1 lattices

This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by K\"{a}mmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve the optimal convergence rate for the $L_{\infty}$ error in Wiener-type spaces, up to logarithmic factors. While this result was translated to weighted Korobov spaces in the recent monograph by Dick, Kritzer, and Pillichshammer (2022), the analysis requires the smoothness parameter $\alpha$ to be greater than $1$ and is restricted to product weights. In this paper, we extend this result for multiple rank-1 lattice-based algorithms to the case where $1/2<\alpha\le 1$ and for general weights, covering a broader range of periodic functions with low smoothness and general relative importance of variables. We also provide a summability condition on the weights to ensure strong polynomial tractability for any $\alpha>1/2$. Furthermore, by incorporating random shifts into multiple rank-1 lattice-based algorithms, we prove that the resulting randomized algorithm achieves a nearly optimal convergence rate in terms of the worst-case root mean squared $L_2$ error, while retaining the same tractability property.