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arXiv · 2512.23001

Around the Fej\'er-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations

Sergey Sadov
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Abstract

The error of approximation of the $2\pi$-periodic sawtooth function $(\pi-x)/2$, $0\leq x<2\pi$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$.

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Sergey Sadov. 2025-12-28. Around the Fej\'er-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations. https://arxiv.org/abs/2512.23001

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