Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 78 / 4 / c

For $f,g\in L^2(0,1)$, interchange the integrals in the inner product of the Volterra integration operator:Thus the adjoint operator is $A^{\ast }g(t)={\int }_t^{1}g(x)dx$. The triangular integration kernel is square-integrable, so $A$ is a Hilbert-Schmidt operator and hence a compact operator.

Set $a_n=(n-\frac{1}{2})\pi$ and ${\sigma }_n=1/a_n$. Direct integration givessince $\cos a_n=0$. Product-to-sum identities give $⟨u_n,u_j⟩=⟨v_n,v_j⟩={\delta }_{nj}$. For completeness, $g=A^{\ast }Af$ satisfies $g^{\prime \prime }=-f$, $g^{\prime }(0)=0$, $g(1)=0$. The associated Sturm-Liouville problem has precisely the complete mixed-boundary cosine basis $u_n$; $A{A}^{\ast }$ gives the mixed-boundary sine basis $v_n$, with $v_n(0)=0$, $v_n^{\prime }(1)=0$. Both $A$ and $A^{\ast }$ are injective, since differentiation of their zero integrals recovers their inputs. Their singular vectors therefore cover the entire two Hilbert spaces, rather than just proper orthogonal complements.

The given functions form the complete singular value system, with ${\sigma }_n=2/((2n-1)\pi )$. This is the mixed-boundary singular system of the Volterra operator.

Substitution into the Tikhonov regularization series gives an explicit reconstruction for every $y\in L^2(0,1)$:Equivalently the summand is ${\sigma }_n⟨y,v_n⟩u_n/(\alpha +{\sigma }_n^2)$. The Tikhonov stability bound makes the series converge in $L^2$ for every positive $\alpha$. It suppresses high-frequency components of unstable differentiation, rather than differentiating arbitrary noisy data pointwise.

The range of the Volterra integration operator is $\{y\in H^1(0,1):y(0)=0\}$. On exactly these data, $A^{†}y=y^{\prime }$ and the series converges to $y^{\prime }$ as $\alpha \downarrow 0$. For arbitrary $L^2$ data outside that range, a finite regularized solution still exists, but a finite-norm unregularized solution need not exist. This is Spectral Tikhonov differentiation for the Volterra operator.