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

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 ${\sigma }_{n-1}$.

For a trigonometric polynomial $t_n(x)={\sum }_{|k|\le n}{c}_k{e}^{ikx}$, orthogonality of the exponential Fourier modes gives $s_j(t_n)=t_n$ whenever $j\ge n$. Every term in the de la Vallée Poussin sum has such an index, soHere and below $n\ge 1$.

Subtracting the two initial averages of Fourier partial sums givesbecause $2n{\sigma }_{2n-1}={\sum }_{j=0}^{2n-1}{s}_j$ and $n{\sigma }_{n-1}={\sum }_{j=0}^{n-1}{s}_j$. The positive normalized Fejér kernel gives the uniform-norm contraction of Fejér summation, namely $\Vert {\sigma }_{m}f{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }$. The triangle inequality therefore gives

For any trigonometric polynomial $t_n$ of degree at most $n$, reproduction from (a) and the operator norm bound from (b) implyTaking the infimum over $t_n$ givesThis is the polynomial reproduction error bound with $\Vert v_n\Vert \le 3$; attaining the best uniform approximation is not needed for this inequality.