# Galerkin Method with Trigonometric Basis on Stable Numerical Differentiation

## TL;DR

This paper develops a Galerkin method with trigonometric basis for stable numerical differentiation of functions on (0, 2π), providing optimal convergence rates, noise-independent parameter choice, and effective handling of discontinuities through numerical examples.

## Contribution

The paper introduces a novel Galerkin approach using trigonometric basis for numerical differentiation, achieving optimal convergence and robustness to noise, with explicit parameter selection strategies.

## Key findings

- Achieves optimal convergence rate of O(δ^{2μ/(2μ+1)})
- Provides noise-independent parameter choice for specific function forms
- Demonstrates effective differentiation of functions with discontinuities

## Abstract

This paper considers the $ p $ ($ p=1,2,3 $) order numerical differentiation on function $ y $ in $ (0,2\pi) $. They are transformed into corresponding Fredholm integral equation of the first kind. Computational schemes with analytic solution formulas are designed using Galerkin method on trigonometric basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2. Therefore, the algorithm achieves the optimal convergence rate $ O( \delta^{\frac{2\mu}{2\mu+1}} ) \ (\mu = \frac{1}{2} \ \textrm{or} \ 1)$ with periodic Sobolev source condition of order $ 2\mu p $. Besides, we indicate a noise-independent a-priori parameter choice when the function $ y $ possesses the form of \begin{equation*} \sum^{p-1}_{k=0} a_k t^k + \sum^{N_1}_{k=1} b_k \cos k t + \sum^{N_2}_{k=1} c_k \sin k t, \ b_{N_1}, c_{N_2} \neq 0, \end{equation*} In particular, in numerical differentiations for functions above, good filtering effect (error approaches 0) is displayed with corresponding parameter choice. In addition, several numerical examples are given to show that even derivatives with discontinuity can be recovered well.