We present a computationally efficient algorithm for stable numerical differentiation from noisy, uniformly-sampled data on a bounded interval. The method combines multi-interval Fourier extension approximations with an adaptive domain partitioning strategy: a global precomputation of local Fourier sampling matrices and their thin SVDs is reused throughout a recursive bisection procedure that selects locally-resolved Fourier fits. Each accepted subinterval stores a compact set of Fourier coefficients that are subsequently used to reconstruct the derivative via a precomputed differentiation operator. The stopping criterion balances fitting error and an explicit noise-level bound, and the algorithm automatically refines the partition where the function exhibits rapid oscillations or boundary activity. Numerical experiments demonstrate significant improvements over existing methods, achieving accurate derivative reconstruction for challenging functions. The approach provides a robust framework for ill-posed differentiation problems while maintaining computational efficiency.

翻译：本文提出一种计算高效的算法，用于在有界区间上从含噪声的均匀采样数据中稳定地进行数值微分。该方法将多区间傅里叶延拓逼近与自适应区域划分策略相结合：在递归二分过程中，通过复用预先计算的局部傅里叶采样矩阵及其薄SVD分解，选取局部可解析的傅里叶拟合。每个被接受的子区间存储一组紧凑的傅里叶系数，随后通过预计算的微分算子重构导数。停止准则在拟合误差与显式噪声水平界之间取得平衡，算法能自动在函数呈现快速振荡或边界活跃的区域细化划分。数值实验表明，该方法相比现有方法有显著改进，能够对挑战性函数实现精确的导数重构。该方案为不适定微分问题提供了一个鲁棒的计算框架，同时保持了计算效率。