Fast Algorithms for Fourier extension based on boundary interval data

TL;DR

This paper introduces a boundary-optimized Fourier extension algorithm that efficiently approximates non-periodic functions with minimal boundary data, achieving high accuracy and reduced boundary oscillation artifacts.

Contribution

It develops a novel boundary-based extension method with a parameter optimization framework, improving convergence and reducing boundary oscillations compared to existing techniques.

Findings

Achieves superalgebraic convergence with few boundary nodes.

Reduces boundary oscillations through grid refinement near boundaries.

Maintains computational complexity close to standard FFT.

Abstract

This paper presents a novel boundary-optimized fast Fourier extension algorithm for efficient approximation of non-periodic functions. The proposed methodology constructs periodic extensions through strategic utilization of boundary interval data, which is subsequently combined with original function samples to form an extended periodic representation. We develop a parameter optimization framework that preserves superalgebraic convergence while requiring only a few boundary node deployment, resulting in computational complexity marginally exceeding that of standard FFT implementations. Furthermore, we present an improved version of the algorithm tailored for functions exhibiting boundary oscillations. This variant employs grid refinement near the boundaries, which reduces the resolution constant to approximately one-fourth of that in conventional approaches. Comprehensive numerical experiments confirm the efficiency and accuracy of the proposed methods and establish practical guidelines for parameter selection.