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

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