In this paper, we first propose a new algorithm for the computation of Fourier extension based on boundary data, which can obtain a super-algebraic convergent Fourier approximation for non-periodic functions. The algorithm calculates the extended part using the boundary interval data and then combines it with the original data to form the data of the extended function within a period. By testing the key parameters involved, their influences on the algorithm was clarified and an optimization setting scheme for the parameters was proposed. Compared with FFT, the algorithm only needs to increase the computational complexity by a small amount. Then, an improved algorithm for the boundary oscillation function is proposed. By refining the boundary grid, the resolution constant of the boundary oscillation function was reduced to approximately 1/4 of the original method.

翻译：本文首先提出了一种基于边界数据计算傅里叶延拓的新算法，该算法能够为非周期函数获得超代数收敛的傅里叶逼近。该算法利用边界区间数据计算延拓部分，随后将其与原始数据结合，构成一个周期内延拓函数的数据。通过对所涉及关键参数的测试，阐明了它们对算法的影响，并提出了一种参数优化设置方案。与快速傅里叶变换（FFT）相比，该算法仅需少量增加计算复杂度。随后，针对边界振荡函数提出了一种改进算法。通过细化边界网格，将边界振荡函数的分辨率常数降至原方法的约四分之一。