Before revision:  Different from Chebyshev approximation, the usual L^\infty-estimate is non-optimal for functions with interior singularities. However, we show that the optimality can be achieved in a certain weightedL^\infty-sense. We also provide point-wise error estimates that can answer some open questions posed in several recent literature.
After revision:  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 several recent literature.