= Endpoint concentration in weighted constant approximation
{title2=$\bar p_0(n)=(2n+1)/(2n+3)\to1$}
The constant minimizing $\int_{-1}^1x^{2n}(x^2-c)^2dx$ is the weighted mean
$$
c=\frac{\int_{-1}^1x^{2n+2}dx}{\int_{-1}^1x^{2n}dx}=\frac{2n+1}{2n+3}.
$$
This 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.
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