Polynomial degree reduction in the ${\mathcal{L}}^2$-norm on a symmetric interval for the canonical basis

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.