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 Kämmerer 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 $α$ 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 $1/2<α\le 1$ 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 $α>1/2$. 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.

💡 Research Summary

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This paper investigates the multivariate approximation of periodic functions belonging to weighted Korobov spaces (H_{d,\alpha,\gamma}) by means of multiple rank‑1 lattice rules. The weighted Korobov space is a reproducing‑kernel Hilbert space characterized by a smoothness parameter (\alpha>1/2) and a collection of non‑negative weights (\gamma={\gamma_u}{u\subset\mathbb N,|u|<\infty}) that quantify the relative importance of variable subsets. Earlier results (Kämmerer & Volkmer 2019; Dick, Kritzer & Pillichshammer 2022) established optimal (L\infty) convergence for (\alpha>1) and product weights only. The present work removes both restrictions: it treats the low‑smoothness regime (1/2<\alpha\le 1) and allows completely general weights (including product, POD and SPOD families).

The authors first recall the classical single‑lattice approach, where the Fourier coefficients of the truncated series are approximated by discrete Fourier transforms on a single rank‑1 lattice. They explain why aliasing—caused by non‑zero dual‑lattice vectors—prevents this method from achieving the optimal rate (N^{-\alpha}) (the best possible for Korobov spaces). The aliasing error forces a convergence ceiling of order (N^{-\alpha/2}) in the deterministic setting.

To overcome this limitation, the paper adopts the multiple‑lattice strategy introduced in earlier works on Wiener‑type spaces. The index set for the truncated Fourier series is chosen as a weighted hyperbolic cross \