In this paper, we develop a direct formula for determining the coefficients in the canonical basis of the best polynomial of degree $M$ that approximates a polynomial of degree $N>M$ on a symmetric interval for the $\mathcal{L}^2$-norm. We also formally prove that using the formula is more computationally efficient than using a classical matrix multiplication approach and we provide an example to illustrate that it is more numerically stable than the classical approach.

翻译：在本文中,我们制定了一个直接公式,用以确定在最佳多元度(M美元)的罐头基数的系数,该公式在对称间隔范围内,在美元\ mathcal{L ⁇ 2$-norm上接近一多边度(N>M美元)。 我们还正式证明,使用公式比使用传统的矩阵乘法在计算上更有效,我们提供了一个例子,说明它比传统方法在数字上更加稳定。