Further Improvements to the Lower Bound for an Autoconvolution Inequality

TL;DR

This paper improves the lower bound for an autoconvolution inequality by constructing a specific step function and refining it through upsampling, achieving a new bound that narrows the gap towards the theoretical maximum.

Contribution

The authors develop a novel step function construction and an upsampling method to significantly improve the known lower bound for the autoconvolution inequality.

Findings

New lower bound of 0.94136 for the autoconvolution inequality.

Constructed a 2,399-interval step function with a bound of 0.926529.

Reduced the gap between the previous bound and the trivial upper limit by roughly 40%.

Abstract

We construct a nonnegative step function comprising 2,399 equally spaced intervals such that \[ \frac{\|f * f\|_{L^{2}(\mathbb{R})}^{2}}{\|f * f\|_{L^{\infty}(\mathbb{R})}\,\|f * f\|_{L^{1}(\mathbb{R})}} \;\ge\; .926529. \] Using a 4x upsampling procedure on this 559-interval optimizer, we further increase the bound to $.94136$, closing roughly 40\% of the gap between the previous best bound (.901562 on 575 intervals) and the trivial upper limit of 1.