Improved bounds on the supremum of autoconvolutions

TL;DR

This paper improves the lower bounds on the maximum value of autoconvolutions of nonnegative functions supported on a compact interval and disproves a longstanding conjecture about their extremal functions.

Contribution

It provides a slight improvement on the lower bounds and offers explicit counterexamples to a well-known conjecture in the field.

Findings

Improved lower bounds for autoconvolution supremum

Counterexamples disproving Schinzel and Schmidt's conjecture

Enhanced understanding of extremal autoconvolution functions

Abstract

We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions.