Periodic functions with bounded remainder

Abstract.

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

$$ \sum\limits_{n = 0}^{N - 1} {f(x_n )} for f \in F $$

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.