  In this paper we present a new perspective on error analysis of Legendre
approximations for differentiable functions. We start by introducing a sequence
of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties,
such as an explicit and optimal upper bound. We then apply these properties to
derive a new and explicit bound for the Legendre coefficients of differentiable
functions and establish some explicit and optimal error bounds for Legendre
projections in the $L^2$ and $L^{\infty}$ norms. Illustrative examples are
provided to demonstrate the sharpness of our new results.
