Periodic functions with bounded remainder
The authors consider sums of the form \(\sum_{n=0}^{N-1} f(x_n)\), where \(f\) is a one-periodic function satisfying certain smoothness conditions and \((x_n)_{n\geq 0}\) is the van der Corput sequence. The main result in the paper is as follows. Let \(f:\mathbb R\to\mathbb R\) be a one-periodic function with an absolutely convergent Fourier series expansion with \(\left|\hat{f}(h)\right|\ll \left| h\right|^{-\alpha}\) for \(\alpha> 1\) and all Fourier coefficients \(\hat{f}(h)\) with \(h\neq 0\). Furthermore, assume that \(\int_0^1 f(x)\,dx=0\). For an integer \(p\geq 2\), let \((x_n)_{n\geq 0}\) be the van der Corput sequence in base \(p\). Then it is true that \[ \left|\sum_{n=0}^{N-1} f(x_n) \right| \ll 1. \] Furthermore, it is shown that this result need not be true if \(\alpha=1\).
scientific article; zbMATH DE number 5126585