A <singular system of a compact operator> is a family $(\sigma_i,u_i,v_i)$ with $\sigma_i>0$, $\sigma_i\to0$, orthonormal $u_i\in Y$ and $v_i\in X$, and
$$
Av_i=\sigma_i u_i,
\qquad
A^*u_i=\sigma_i v_i.
$$
On each singular vector, $A^*A+\alpha I$ acts by multiplication by $\sigma_i^2+\alpha$. Hence
$$
\boxed{
x_\alpha^\delta
=\sum_{i=1}^\infty
\frac{\sigma_i}{\sigma_i^2+\alpha}
(y^\delta,u_i)v_i.}
$$
The bounded filter $\sigma/(\sigma^2+\alpha)$ replaces the unstable inverse factor $1/\sigma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 3 / ii / Solution

A singular system $(\sigma_j,u_j,v_j)$ for a compact operator $A:X\to Y$ has positive singular values $\sigma_j\to0$ and orthonormal vectors satisfying $Av_j=\sigma_j u_j$ and $A^*u_j=\sigma_jv_j$. It diagonalizes spectral regularization methods.


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