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

TL;DR

This paper establishes tight bounds for approximation errors of certain oscillatory functions using Laplace transform representations, extending classical inequalities with explicit constants and asymptotic sharpness.

Contribution

It introduces new tight bounds for approximation errors of oscillatory functions, utilizing Laplace transform representations of the Lerch Zeta function, and extends Fejér-Jackson inequalities.

Findings

Bound on Fourier approximation error of the sawtooth function.

Asymptotically tight inequalities with explicit constants.

Results for Dirichlet kernel interpolation and Taylor series remainders.

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 $s=1$.