 Mixed-boundary singular system of the Volterra operator

For the Volterra integration operator $Af(x)={\int }_0^{x}f(t)dt$, its adjoint is $A^{\ast }g(x)={\int }_x^{1}g(t)dt$. With $a_n=(n-\frac{1}{2})\pi$, the input singular vectors are $\sqrt{2}\cos (a_{n}x)$ and the output singular vectors $\sqrt{2}\sin (a_{n}x)$, 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.