= Mixed-boundary singular system of the Volterra operator
{title2=$\sigma_n=((n-\tfrac12)\pi)^{-1}$}
For the <Volterra integration operator> $Af(x)=\int_0^x f(t)dt$, its adjoint is $A^*g(x)=\int_x^1g(t)dt$. With $a_n=(n-\tfrac12)\pi$, the input singular vectors are $\sqrt2\cos(a_nx)$ and the output singular vectors $\sqrt2\sin(a_nx)$, with singular values $1/a_n$. The associated <Sturm-Liouville problem> has Neumann condition at zero and Dirichlet condition at one for the cosine basis; the sine basis has the reversed conditions. These complete <orthonormal bases> give a <singular value system> for an injective <compact operator> with dense range.
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