Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 67 / 2

For a continuous $2\pi$-periodic function, define the Fourier coefficients and Fourier partial sum byThe Fejér sum is the arithmetic average of the first $n$ Fourier partial sums:Thus $s_n$ has degree at most $n$, whereas ${\sigma }_n$ has degree at most $n-1$. These indexing conventions matter in the identity in (b). Throughout, $m$ is a positive integer; when $n=0$, any term $n{\sigma }_n$ is interpreted as zero without introducing ${\sigma }_0$.

For later estimates, the Fejér kernel in the PDF's normalization isThe apparent singularities are removable. The squared modulus makes $F_n$ nonnegative, and integrating its finite exponential expansion leaves only the $n$ diagonal terms, so ${\pi }^{-1}\int F_n=1$. In particular Fejér summation is a uniform-norm contraction:

A trigonometric polynomial $t_n$ of degree at most $n$ has zero Fourier coefficients outside $[-n,n]$. Its Fourier partial sum $s_j(t_n)$ therefore equals $t_n$ for every $j\ge n$. Averaging the sums with $j=n,\dots ,n+m-1$ givesThus the de la Vallée Poussin sum reproduces the entire degree-at-most-$n$ approximation space, including when the actual degree is smaller.

The difference between the two cumulative sums of Fourier partial sums isHence the de la Vallée Poussin sum satisfiesUsing the uniform-norm contraction of Fejér summation on each term and the triangle inequality in the supremum norm,This proves the requested operator norm bound. When $n=0$ the identity reduces to $v_{0,m}={\sigma }_m$ and the bound is one.

Take any trigonometric polynomial $t\in {\mathcal{T}}_n$. The reproduction property in (a) givesBy (b),Taking the infimum over $t\in {\mathcal{T}}_n$ yields the polynomial reproduction error bound for this operator. With $E_n(f)={\inf }_{t\in {\mathcal{T}}_n}\Vert f-t{\Vert }_{\infty }$ and $n/m\le M$, it becomesThis is a near-best uniform approximation estimate; it follows from reproduction and boundedness without requiring the de la Vallée Poussin sum itself to be a best approximant.