A Fejér type extremal problem (Q1187272)

Let \[ {\mathcal F}_ k=\left\{f; \forall x f(x)\geq 0, f(x)=1+a_ 1 \cos x+\sum^ \infty_{n=k+1} a_ n \cos nx\right\} \] and \(\alpha(k)=\sup\{a_ 1; f\in{\mathcal F}_ k\}\). The author gives the following sharp estimation of \(\alpha(k)\), namely \(1/\cos{\pi\over k+2}\).