Endpoint concentration in weighted constant approximation (${\bar{p}}_0(n)=(2n+1)/(2n+3)\to 1$)

The constant minimizing ${\int }_{-1}^1{x}^{2n}(x^2-c{)}^{2}dx$ is the weighted meanThis follows by differentiating the quadratic error or requiring the residual to be orthogonal to constants. The normalized weight $(2n+1)x^{2n}/2$ assigns mass $r^{2n+1}$ to $[-r,r]$ for every $0<r<1$. As $n$ increases it concentrates at the endpoints, where $x^2=1$, explaining the limiting best constant. Although the weight vanishes at zero, its integral against the square of a nonzero continuous function is positive, so the associated weighted inner product is positive definite.