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

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