Resumen de Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials
We obtain optimal trigonometric polynomials of a given degree N that majorize, minorize and approximate in 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.
