Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 67 / 2 / c

Table of contents
- Solution c

Solution

Take any trigonometric polynomial

t \in T_{n}

. The reproduction property in (a) gives

f - v_{n, m} f = (f - t) - v_{n, m} (f - t) .

By (b),

∥ f - v_{n, m} f ∥_{\infty} \leq (2 + \frac{2 n}{m}) ∥ f - t ∥_{\infty} .

Taking the infimum over

t \in T_{n}

yields the polynomial reproduction error bound for this operator. With

E_{n} (f) = in f_{t \in T_{n}} ∥ f - t ∥_{\infty}

and

n / m \leq M

, it becomes

∥ f - v_{n, m} f ∥_{\infty} \leq 2 (M + 1) E_{n} (f) .

This is a near-best uniform approximation estimate; it follows from reproduction and boundedness without requiring the de la Vallée Poussin sum itself to be a best approximant.

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