Periodic functions with bounded remainder

Let F be the class of all 1-periodic real functions with absolutely convergent Fourier series expansion and let (x_n)_n ≧ 0 be the van der Corput sequence. In this paper results on the boundedness of 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. The first author is supported by the Australian Research Council under its Center of Excellence Program. The second author is supported by the Austrian Research Foundation (FWF), Project S9609 that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. Received: 14 December 2005