Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 11F / b / v

Table of contents
- Solution v

Solution

True. By the Weierstrass approximation theorem, choose polynomials

q_{n}

with

q_{n} \to f^{(m)}

uniformly. Define

P_{n}

by integrating

q_{n}

exactly

m

times and choose the

m

integration constants so that

P_{n}^{(r)} (0) = f^{(r)} (0), 0 \leq r < m .

Then

P_{n}^{(m)} = q_{n}

, and repeated use of the fundamental theorem of calculus gives

∥ P_{n}^{(r)} - f^{(r)} ∥_{\infty} \leq \frac{∥ q _{n} - f ^{(m)} ∥ _{\infty}}{( m - r )!} (0 \leq r < m) .

Every derivative therefore converges uniformly as required.

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