Periodic functions with bounded remainder

Let F be the class of all 1-periodic real functions with absolutely convergent Fourier series expansion and let (xn)n≥0 be the van der Corput sequence. In this paper results on the boundedness of ∑N−1 n=0 f(xn) for f ∈ F are given. We give a criterion on the convergence rate of the Fourier coefficients of f such that the above sum is bounded independently of N . Further we show that our result is also best possible.