  We obtain optimal trigonometric polynomials of a given degree $N$ that
majorize, minorize and approximate in $L^1(\mathbb{R}/\mathbb{Z})$ the
Bernoulli periodic functions. These are the periodic analogues of two works of
F. Littmann that generalize a paper of J. Vaaler. As applications we provide
the corresponding Erd\"{o}s-Tur\'{a}n-type inequalities, approximations to
other periodic functions and bounds for certain Hermitian forms.
