In this paper, new relations between the derivatives of the Legendre polynomials are obtained, and by these relations, new upper bounds for the Legendre coefficients of differentiable functions are presented. These upper bounds are sharp and cover more categories of differentiable functions. Moreover, new and sharper bounds for the approximation error of the partial sums of Legendre polynomials are provided. Numerical examples are given to validate our theoretical results.

翻译：在本文中,图例多义的衍生物之间有了新的关系,通过这些关系,提出了不同功能的图例系数的新上限。这些上限很尖锐,覆盖了更多的不同功能类别。此外,还提供了图例多义部分金额近似差错的新和较清晰的界限。提供了数字例子来验证我们的理论结果。