Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 5 / 3 / ii / a

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- Solution a

Solution

The domination hypothesis includes

r = 0

. For any approximation error

h

, linearity and the triangle inequality therefore give the uniform estimate

∣ T_{r} h - T_{0} h ∣ \leq ∣ T_{r} h ∣ + ∣ T_{0} h ∣ \leq 2 S (h) .

For a general input

g

and an approximating

f

, this implies

∣ T_{r} g - T_{0} g ∣ \leq ∣ T_{r} f - T_{0} f ∣ + 2 S (g - f) .

The first term converges on the dense class, while the weak type (1,1) estimate controls the exceptional set where the second is large.

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