= Solution
Averaging the given <Dirichlet kernels> gives the normalized <Fejér kernel>
$$
K_n(t)=\frac1{\pi n}\sum_{j=0}^{n-1}D_j(t)
=\frac1{2\pi n}\left(\frac{\sin(nt/2)}{\sin(t/2)}\right)^2.
$$
The geometric-series identity, or summing the sine terms, proves the second equality. At multiples of $2\pi$ use its removable value $n/(2\pi)$. Therefore $K_n\geq0$, and
$$
\sigma_n(f,x)=\int_{-\pi}^{\pi}K_n(t)f(x-t)\,dt
$$
is a <positive linear operator on continuous functions>. The <Fourier partial sums>, viewed as <orthogonal projections>, fix $1$, so $\sigma_n1=1$ and $\int K_n=1$. They also fix $\sin x$ and $\cos x$ for degrees at least one, whereas $s_0$ annihilates them. Consequently
$$
\sigma_n(\sin x)=(1-1/n)\sin x,\qquad
\sigma_n(\cos x)=(1-1/n)\cos x.
$$
All three periodic <Korovkin theorem> tests converge uniformly. Hence \b[<Fejér sums> converge uniformly for every continuous periodic function]:
$$
\boxed{\|\sigma_n(f)-f\|_\infty\longrightarrow0.}
$$
Back to article page
