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：在本文中,我们提出了对不同功能图例近似值错误分析的新观点。 我们首先引入了图例- Gaus- Lobatto 多边协议的序列,并证明了它们的理论特性,例如一个明确和最佳的上限。 然后,我们运用这些属性来为不同功能的图例系数产生一个新的和明确的约束,并为图例预测设定一些明确和最佳的错误界限,即$L2$和$L ⁇ infty} 规范。我们提供了一些说明性的例子,以表明我们新结果的清晰性。