 Admissible polynomials for a singular inverse-linear weight

Consider the squared error of approximating $x^n$, $n\ge 1$, by a polynomial $q$ under a measure containing $dx/x$ on $(0,1)$. This error is finite exactly when $q(0)=0$. A nonzero constant term makes the integrand asymptotic to $q(0{)}^2/x$, which is not Lebesgue integrable; a vanishing constant term makes $x^n-q(x)=O(x)$, giving an integrable $O(x)$ integrand.

Consequently the full space of polynomials of degree at most $n-1$ is not contained in the weighted L2 space. Its admissible subspace is $\mathrm{span}\{x,\dots ,x^{n-1}\}$, or zero when $n=1$. This finite-dimensional subspace has a unique orthogonal projection of $x^n$, hence a unique best approximation. Indeed its Gram matrix is positive definite: a nonzero polynomial cannot vanish on an interval, so its squared norm is positive. Thus a singular weight can invalidate the proposed full approximation space while still allowing a unique finite-error minimizer in its admissible part. As usual, weighted L2 spaces identify functions equal almost everywhere.