A lattice algorithm with multiple shifts for function approximation in Korobov spaces

Abstract

In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ $\mathcal{O}((\log N)^{d} )$ good shifts and recover each Fourier coefficient via a least-squares procedure. We show that the resulting approximation achieves the optimal convergence rate for the $L_{\infty}$-approximation error in the worst-case setting, namely $\mathcal{O}(N^{-\alpha+1/2+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Moreover, by incorporating random shifts, the algorithm attains the optimal rate for the $L_{2}$-approximation error in the randomized setting, which is $\mathcal{O}(N^{-\alpha+\varepsilon})$. Numerical experiments are presented to support the theoretical results.