Error analysis of truncation Legendre method for solving numerical differentiation

We study the problem of numerical differentiation of functions from weighted Wiener classes. This problem is fundamental in computational mathematics and presents significant challenges due to its ill-posed nature, where small perturbations in input data can lead to substantial errors in derivative approximations. We construct and analyze a truncation Legendre method to recover arbitrary order derivatives, extending beyond the traditional focus on first-order derivatives. The method utilizes the orthogonal properties of Legendre polynomials and employs a truncation parameter that balances the approximation error with the propagation of input data perturbations. Our main contribution is obtaining explicit error estimates in both integral and uniform metrics, significantly generalizing previous results that were limited to specific cases. We establish optimal choices for the truncation parameter as a function of the noise level, smoothness parameter, and the order of derivative. The analysis covers a comprehensive range of parameters characterizing the function class, derivative order, and metrics of both input and output spaces. We prove the accuracy of the proposed method in the weighted Wiener class, providing explicit convergence rates. Our theoretical framework enables practitioners to determine the optimal balance between computational complexity and accuracy for their specific applications.