Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 61 / 4 / c

For the indicated function the cosine substitution givesThe absolute-value function is Lipschitz continuous with constant one, and so is sine. Their composition therefore satisfiesUse the sharper intermediate estimate from the preceding part, rather than the ordinary interval modulus of continuity:

The usual inverse theorem for trigonometric approximation cannot hold verbatim with the ordinary interval modulus of continuity. It would implyBut comparison with the endpoint $1$ giveswhich is of order $n^{-1/2}$ and contradicts that bound as $n\to \infty$. This is an endpoint obstruction to an algebraic inverse approximation theorem. The algebraic approximation rate measures smoothness after cosine substitution; cosine compresses distances quadratically near the endpoints. A valid algebraic inverse theorem must account for that endpoint geometry rather than using the unchanged periodic formulation.