= Shifted Hilbert quadratic-form bound
{title2=$0\leq\sum_{r,s=1}^n\frac{a_ra_s}{r+s+1}\leq\pi\sum_{r=1}^na_r^2$}
For real $a_r$, put $h(z)=\sum_{r=1}^na_rz^r$. The quadratic form equals $\int_0^1h(x)^2dx$, proving its nonnegativity. Apply the <centred logarithmic keyhole identity> bound to the <entire function> $h^2$. Since $|h(e^{i\theta})^2|=|h(e^{i\theta})|^2$, orthogonality of the finite Fourier sum gives $\int_0^{2\pi}|h(e^{i\theta})|^2d\theta=2\pi\sum_ra_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a <Hilbert matrix> obtained by deleting its first row and column.
Back to article page
