Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence

TL;DR

This paper reveals a surprising coincidence between the Lebesgue constants of the Walsh system and the star discrepancy of the van der Corput sequence, linking approximation theory and uniform distribution theory.

Contribution

It demonstrates that these two previously studied quantities are actually equal, unifying results and insights from both mathematical areas.

Findings

Lebesgue constants and star discrepancy are equal.

Many theorems are common to both areas.

Some results were previously known only in one context.

Abstract

In this short note we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important role in approximation theory, and on the other hand there is the star discrepancy of the van der Corput sequence that plays a prominent role in uniform distribution theory. Over the decades, these two quantities have been examined in great detail independently of each other and important results have been proven. Work in these areas has been carried out independently, but as we show here, they actually coincide. Interestingly, many theorems have been discovered in both areas independently, but some results have only been known in one area but not in the other.