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.

翻译：本文提出了可微函数勒让德逼近误差分析的新视角。我们首先引入勒让德-高斯-洛巴托多项式序列，并证明其理论性质，例如显式且最优的上界。随后应用这些性质推导可微函数勒让德系数的显式新界，并建立$L^2$和$L^{\infty}$范数下勒让德投影的若干显式最优误差界。通过示例验证了新结果的精确性。