标题： 关于三角插值及其应用 显示英文标题

标题： On Trigonometric Interpolation and Its Applications

摘要： 本文提出了一种新的三角插值算法，并建立了相关的收敛性质。该方法调整了一个现有的三角插值算法，使其能够更好地利用快速傅里叶变换（FFT）来提高效率。该算法可以被制定出来，以便有效地利用某些抵消效应来进行误差分析，这不仅使我们能够获得函数逼近所需的均匀收敛率，还能得到其导数的所需均匀收敛率。我们进一步改进了该算法，使其可以应用于定义在有界区间上的非周期函数。数值测试结果证实了该算法具有良好的准确性能。在应用方面，我们展示了如何将其应用于估计积分和求解线性/非线性常微分方程（ODE）。测试结果显示，它在积分上显著优于梯形/辛普森方法，在常微分方程上显著优于标准龙格-库塔算法。此外，我们还展示了一些数值证据表明该算法的估计误差可能表现出“局部特性”，即一个点的误差倾向于不会传播，从而避免在其他地方产生显著的误差累积，这是与基于多项式的近似相比的一个显著优势。 显示英文摘要

摘要： In this paper, we propose a new trigonometric interpolation algorithm and establish relevant convergent properties. The method adjusts an existing trigonometric interpolation algorithm such that it can better leverage Fast Fourier Transform (FFT) to enhance efficiency. The algorithm can be formulated in a way such that certain cancellation effects can be effectively leveraged for error analysis, which enables us not only to obtain the desired uniform convergent rate of the approximation to a function, but desired uniform convergent rates for its derivatives as well. We further enhance the algorithm so it can be applied to non-periodic functions defined on bounded intervals. Numerical testing results confirm decent accurate performance of the algorithm. For its application, we demonstrate how it can be applied to estimate integrals and solve linear/non-linear ordinary differential equation (ODE). The test results show that it significantly outperforms Trapezoid/Simpson method on integral and standard Runge-Kutta algorithm on ODE. In addition, we show some numerical evidences that estimation error of the algorithm likely exhibits ``local property", i.e. error at a point tends not to propagate, which avoids significant compounding error at some other place, as a remarkable advantage compared to polynomial-based approximations.