Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 2 / 1 / Solution

Use degree-indexed Fourier partial sums and Fejér sums, as in this paper. For a $2\pi$-periodic continuous function, putEquivalently,These are trigonometric polynomials of degree at most $n$. Some treatments index the Fejér sum by the number of terms rather than its degree; the degree convention here is why the formulas below contain ${\sigma }_{2n-1}$ and ${\sigma }_{n-1}$.