Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / c

Table of contents
- Solution c

Solution

Twice applying the fundamental theorem of calculus gives

f (x + h) - 2 f (x) + f (x - h) = \int_{0}^{h} \int_{- s}^{s} f^{''} (x + u) d u d s,

hence

ω_{2} (f, t) \leq t^{2} ∥ f^{''} ∥_{\infty}

. Part (b) gives

∥ σ_{n} (f) - f ∥_{\infty} = O (n^{- 1})

. This cannot be little-

o

for every

C^{2}

f_{0} (x) = cos x

,

σ_{n} (f_{0}) = (1 - n^{- 1}) cos x, ∥ σ_{n} (f_{0}) - f_{0} ∥_{\infty} = n^{- 1} .

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