Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence
Josef Dick, Friedrich Pillichshammer
TL;DR
The paper uncovers a deep coincidence between two classical measures: the Lebesgue constants of the Walsh system and the star discrepancy of the van der Corput sequence. It proves the exact equality $d_n = L_n$ with an explicit expression $d_n = ν - ∑_{1 ≤ j < i ≤ ν} 2^{n_i - n_j}$ for $n = ∑ 2^{n_r}$, and shows $L_{2n}=L_n$ and related recurrences and generating functions. This equivalence enables seamless transfer of results between approximation theory and discrepancy theory, yields new representations for both $D_n^{*}$ and $L_n$, and links central limit theorems and asymptotics across the two domains. Overall, it unifies two independent lines of study and expands the toolkit for uniform distribution and Fourier approximation research.
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.