Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 326 / 1 / c

Use the singular system of a compact operator convention $A{x}_j={\sigma }_j{y}_j$ and $A^{\ast }{y}_j={\sigma }_j{x}_j$. Starting from zero gives no component in $\ker A$. Write $x(t)={\sum }_j{c}_j(t)x_j$. The coefficient equations areSolving these scalar linear ordinary differential equations givesConsequently asymptotic regularization has the spectral regularization method representationThe component of $f$ orthogonal to the closure of the range of $A$ is annihilated by $A^{\ast }$ and contributes nothing. The scalar filter satisfies $(1-e^{-{\sigma }^2/\alpha })/\sigma \le \min (\sigma /\alpha ,1/\sigma )\le {\alpha }^{-1/2}$, so every fixed-$\alpha$ reconstruction is a bounded linear operator. Its long-time limit is the Moore–Penrose inverse of an operator on admissible data.