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

Table of contents
- Solution ii

Solution

A singular system of a compact operator is a family

(σ_{i}, u_{i}, v_{i})

with

σ_{i} > 0

,

σ_{i} \to 0

, orthonormal

u_{i} \in Y

and

v_{i} \in X

, and

A v_{i} = σ_{i} u_{i}, A^{*} u_{i} = σ_{i} v_{i} .

On each singular vector,

A^{*} A + α I

acts by multiplication by

σ_{i}^{2} + α

. Hence

x_{α}^{δ} = i = 1 \sum \infty \frac{σ _{i}}{σ _{i}^{2} + α} (y^{δ}, u_{i}) v_{i} .

The bounded filter

σ / (σ^{2} + α)

replaces the unstable inverse factor

1/ σ

.

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