Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations

TL;DR

The paper develops a unified Laplace-transform approach to produce tight, explicit envelopes for oscillatory sums and integrals arising from Dirichlet/Lerch-type series, and applies it to the Taylor remainder of $\log(1-z)$ in the unit disk. Central to the method is the comparison function $M(t)=\int_0^\infty \frac{e^{-tu}}{\sqrt{u^2+1}}\,du$ and Laplace representations that relate Fejér-Jackson sums to integrated Dirichlet kernels, yielding sharp, constants-free bounds. The authors prove a sharpened Fejér-Jackson-Turán inequality $|S(x,0)-S_n(x,0)|\le \operatorname{arccot}((2n+1)\sin\frac{x}{2})$ and develop a suite of related inequalities for FJ sums, integrated kernels, and the logarithmic remainder, with precise $n$-dependent envelopes. These results advance understanding of the Gibbs phenomenon and provide explicit, practical tools for bounding oscillatory phenomena in harmonic analysis and complex analysis contexts.

Abstract

The error of approximation of the $2π$-periodic sawtooth function $(π-x)/2$, $0\leq x<2π$, 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 $s=1$.

Figures (2)

• Figure 1: The functions $\lambda \mathrm{Cci}(x,=\lambda)$ (green), $\lambda \mathrm{Sci}(x,\lambda)-1$ (red), and their envelopes $\pm \sec x$ for $\lambda=12$
• Figure 2: The function $S_{10}(x,0)$, its known minorants and the envelopes \ref{['ineq:myFJT']}

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof