A new and sharper bound for Legendre expansion of differentiable functions

TL;DR

This paper introduces a more precise bound for Legendre coefficients of differentiable functions, leading to improved error estimates for truncated Legendre series, with proofs based on integration by parts and Bernstein inequalities.

Contribution

It presents a novel, sharper bound for Legendre coefficients and an improved error estimate for Legendre series truncation, enhancing approximation accuracy.

Findings

New sharper bound for Legendre coefficients

Improved error bounds for truncated Legendre series

Demonstrated sharpness through illustrative example

Abstract

In this paper, we provide a new and sharper bound for the Legendre coefficients of differentiable functions and then derive a new error bound of the truncated Legendre series in the uniform norm. The key idea of proof relies on integration by parts and a sharp Bernstein-type inequality for the Legendre polynomial. An illustrative example is provided to demonstrate the sharpness of our new results.