Proof of Weierstrass' second theorem using the Fejér operator Weierstrass' second theorem states the following:
Let $f$ be a real continuous $2\pi$-periodic function (write $f\in C_{2\pi}$). Then for all $\epsilon>0$ there exists a trigonometric polynomial $p$ such that $\|f-p\|_{\infty}<\epsilon$
This theorem can be proved using a trigonometric version of Korovkin's lemma with the Fejér operator
$$H_n(f;\theta)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)F_n(t-\theta)dt$$
where
$$F_n(t)=\frac{1}{2n}\frac{\sin^2(\frac{nt}{2})}{\sin^2(\frac{t}{2})}=\frac{1}{2}+\sum\limits_{k=1}^{n-1}\bigg(1-\frac{k}{n}\bigg)\cos(kt)$$
My question is how to show that $H_n(f;\theta)$ is a trigonometric polynomial

$F_n(t)$ is a trigonometric polynomial of "degree" $n-1$, as exhibited by your second formula. Therefore the functions
$$g_t(\theta):= F_n(t-\theta)={1\over2}+\sum_{k=1}^{n-1}\left(1-{k\over n}\right)\bigl(\cos(kt)\cos(k\theta)+\sin(kt)\sin(k\theta)\bigr)$$
are trigonometric polynomials in $\theta$ for each fixed $t$. It follows that the function $$\theta\mapsto H_n(f;\theta)\ ,$$
being a "linear combination" of such $g_t$, is a trigonometric polynomial in $\theta$ of degree $n-1$, whose coefficients $a_k$, $b_k$ are given by certain integrals involving $f$.
