A local Fourier extension method for function approximation

This paper proposes a novel localized Fourier extension method for approximating non-periodic functions via domain segmentation. By partitioning the computational domain into subregions with uniform discretization scales, the method achieves spectral accuracy at $ \mathcal{O}(M) $ computational complexity. Theoretical error bounds and parameter dependency analyses validate the robustness of the proposed method. The relationship among the key parameters involved is analytically established, accompanied by an optimized parameter selection strategy. Numerical experiments further confirm the effectiveness of the proposed method.