Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

We obtain optimal trigonometric polynomials of a given degree NN that majorize, minorize and approximate in L1(R/Z)L1(R/Z) the Bernoulli periodic functions. These are the periodic analogues of two works of Littmann [F. Littmann, Entire majorants via Euler–Maclaurin summation, Trans. Amer. Math. Soc. 358 (7) (2006) 2821–2836; F. Littmann, Entire approximations to the truncated powers, Constr. Approx. 22 (2) (2005) 273–295] that generalize a paper of Vaaler [J.D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985) 183–215]. As applications we provide the corresponding Erdös–Turán-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.