Fast numerical derivatives based on multi-interval Fourier extension

Abstract: 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.