Error Analysis of Truncation Legendre Method for Solving Numerical Differentiation

TL;DR

This paper provides a comprehensive error analysis of a truncation Legendre method for numerical differentiation across various function spaces, establishing optimal convergence rates and truncation parameters.

Contribution

It extends previous work by analyzing the method's error bounds in multiple metrics for a broad class of functions, not just first derivatives.

Findings

Achieves optimal convergence rates in weighted Wiener classes.

Determines optimal truncation parameters based on error and smoothness.

Provides error bounds in C and L_q metrics for 2 ≤ q ≤ ∞.

Abstract

We study the problem of numerical differentiation of functions from weighted Wiener classes. We construct and analyze a truncation Legendre method to recover arbitrary order derivatives. The main focus is on obtaining error estimates in integral and uniform metrics. Unlike previous studies, which predominantly focused on first-order derivatives and specific functional spaces, we conduct a comprehensive analysis across a wide spectrum of function regularity parameters and various metrics for measuring errors. We establish precise error bounds for the truncation method in the metrics of C and L_q for 2 less than or equal to q less than or equal to infinity, and determine optimal truncation parameters as functions of the error level and smoothness parameters. Our results demonstrate that the truncation method achieves optimal convergence rates on weighted Wiener classes, requiring an optimal number of perturbed Fourier-Legendre coefficients for effective derivative recovery.