Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

Details Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

2008-09-23

arxiv.org/abs/0809.4049v2

Journal of Approximation Theory (Print), v. 154, p. 90-104, 2008

Mathematics

Classical Analysis and ODEs

14 pages. Accepted for publication in the J. Approx. Theory. V2 has additional references and some typos corrected

Scientific paper

10.1016/j.jat.2008.03.007

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.

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