Fourier Extension Based on Weighted Generalized Inverse

TL;DR

This work addresses spurious oscillations in Fourier extensions by embedding a priori smoothness into a weighted generalized inverse, forming a compact-operator framework. The authors define a weighted regularization with $ R= B_{h_r}$ and $ G= F R^{-1}$, solving the regularized problem via a generalized truncated SVD to obtain $c^{psilon,ta}$ and recover the extension with controlled oscillations. They prove $L^2$-optimal convergence on $[-1,1]$ and derivative stability under smoothness assumptions, and implement a uniform-sampling discretization with adaptive weight design $h_{r,N}$ to balance accuracy and regularization. Numerical tests show reduced oscillations in the extended region and markedly improved derivative approximations over the classical Fourier extension, demonstrating the approach's practical impact for stable, smooth extensions in applications requiring accurate derivatives.

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.

Figures (5)

• Figure 1: The approximation error $\|f-\mathcal{Q}_{\gamma,N}^{T,{\bf R}}{\bf f}^{\epsilon}\|$ against $T$ for different values of $\omega$, $\gamma$ and $p$.
• Figure 2: Calculation results of $\mathcal{Q}_{\gamma,N}^{T,{\bf R}}{\bf f}^{\epsilon}$ for the various test functions ($N=200$).
• Figure 3: Approximation errors for the various test functions.
• Figure 4: Approximation error of the first-order derivative for various test functions.
• Figure 5: Approximation error of the second-order derivative for various test functions.

Theorems & Definitions (9)

• lemma thmcounterlemma
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• theorem 1
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• corollary thmcountercorollary: Convergence rates under typical smoothness assumptions
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