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A real symmetric matrix whose entries are the moments $H_{ij}={\int }_0^1{x}^{i-1}{x}^{j-1}dx$. Thus it is the Gram matrix of the monomials in the real L2 inner product on $[0,1]$. For a nonzero real coefficient vector $v$, $v^{T}Hv={\int }_0^1({\sum }_i{v}_i{x}^{i-1}{)}^{2}dx>0$, so every finite Hilbert matrix is positive-definite. The moment description connects these matrices with polynomial approximation and with the shifted Hilbert quadratic-form bound.

On the real interval, $h(x)$ is real and $H(x)=h(x{)}^2\ge 0$. Expanding and integrating the finite sum yieldsApply part (b)(iv) to the entire function $H=h^2$. Since $|H(e^{i\theta })|=|h(e^{i\theta }){|}^2$, orthogonality of complex exponentials givesConsequentlyThe quadratic form is actually nonnegative, so its absolute value can be dropped. This shifted Hilbert quadratic-form bound comes from combining the logarithmic contour estimate with a finite Parseval identity.