Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 3 / b / Solution

By the spectral theorem, the operator in parentheses acts at spectral value

t \in R

by

\frac{1}{2 πi} \int_{a}^{b} [\frac{1}{t - x - i ϵ} - \frac{1}{t - x + i ϵ}] d x = \frac{1}{π} \int_{a}^{b} \frac{ϵ}{( t - x ) ^{2} + ϵ ^{2}} d x .

The Poisson kernel converges to

1

for

t \in (a, b)

, to

1/2

at

t = a, b

, and to

0

outside

[a, b]

. It is uniformly bounded, so dominated convergence in the spectral measure of a normal operator gives the strong limit

E ((a, b)) + \frac{1}{2} E ({a}) + \frac{1}{2} E ({b}) = \frac{1}{2} [E ((a, b)) + E ([a, b])] .

Applying this operator to

v

proves the claimed Stone formula.

Solved by gpt-5.6-sol high.

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