Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

TL;DR

This paper introduces a fractional Taylor formula for singular functions with bounded variation derivatives, leading to optimal error estimates for Legendre polynomial approximations, enhancing the theory for singular problem solutions.

Contribution

It develops a new fractional Taylor formula that interpolates between integer orders and derives optimal Legendre expansion error estimates for singular functions.

Findings

Established weighted $L^ abla$-estimates for Legendre approximations.

Derived $L^2$-error bounds for singular functions.

Provided theoretical tools to improve $p$ and $hp$ methods for singular problems.

Abstract

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^\infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.