We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability distributions, with the help of spectral filters. We establish general convergence rates for these filters and further show that several classical spectral filters achieve convergence rates one order faster than previously recognized in the literature on the Gibbs phenomenon. Our numerical experiments corroborate the theoretical convergence results. Additionally, we illustrate the computational speed and accuracy of the discrete COS method with applications in computational statistics and quantitative finance. The theoretical and numerical results highlight the method's potential for solving problems involving discrete distributions, particularly when the characteristic function is known, allowing the discrete Fourier transform (DFT) to be bypassed.

翻译：我们提供了一个严格的收敛性证明，表明借助谱滤波器，著名的用于连续概率分布逆傅里叶变换的半解析傅里叶余弦（COS）公式可以推广到离散概率分布。我们为这些滤波器建立了通用的收敛速率，并进一步证明，在吉布斯现象的文献中，几种经典谱滤波器实现的收敛速率比先前认识到的快一个数量级。我们的数值实验证实了理论收敛结果。此外，我们通过计算统计学和定量金融中的应用，说明了离散COS方法的计算速度和准确性。理论和数值结果突显了该方法在解决涉及离散分布问题方面的潜力，特别是在已知特征函数的情况下，可以绕过离散傅里叶变换（DFT）。