Paper detail

Fourier Extension Based on Weighted Generalized Inverse

This paper introduces a weighted generalized inverse framework for Fourier extensions, designed to suppress spurious oscillations in the extended region while maintaining high approximation accuracy on the original interval. By formulating the Fourier extension problem as a compact operator equation, we propose a weighted best-approximation solution that incorporates a priori smoothness information through suitable weight operators on the Fourier coefficients. This leads to a regularization scheme based on the generalized truncated singular value decomposition (GTSVD). Under algebraic and exponential smoothness assumptions, convergence analysis demonstrates optimal $L^2$ accuracy and improved stability for derivatives. Compared with classical Fourier extension using standard TSVD, the proposed method effectively controls high-frequency components and yields smoother extensions. A practical discretization using uniform sampling is developed, along with an adaptive design of weight functions. Numerical experiments confirm that the method significantly improves derivative approximations and reduces oscillations in the extended domain without compromising accuracy on the original interval.