= Admissible polynomials for a singular inverse-linear weight
Consider the squared error of approximating $x^n$, $n\ge1$, 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 $\operatorname{span}\{x,\ldots,x^{n-1}\}$, 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.
Back to article page
