Fourier Extension Based on Weighted Generalized Inverse

TL;DR

This paper presents a weighted generalized inverse approach for Fourier extensions that reduces oscillations and enhances approximation accuracy by incorporating smoothness information and regularization techniques.

Contribution

It introduces a novel weighted framework using GTSVD for Fourier extensions, improving stability and control over high-frequency oscillations compared to classical methods.

Findings

Enhanced control of oscillations in Fourier extensions.

Improved stability and accuracy for derivatives.

Effective suppression of high-frequency components.

Abstract

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.