On Properties and the Application of Levin-type Sequence Transformations for the Convergence Acceleration of Fourier Series∗

We discuss Levin-type sequence transformations {sn} → {sn} that depend linearly on the sequence elements sn, and nonlinearly on an auxiliary sequence of remainder estimates {ωn}. If the remainder estimates also depend on the sequence elements, non-linear transformations are obtained. The application of such transformations very often yields new sequences that are more rapidly convergent in the case of linearly and logarithmically convergent sequences. Also, divergent power series can often be summed, i.e., transformed to convergent sequences, by such transformations. The case of slowly convergent Fourier series is more difficult and many known sequence transformations are not able to accelerate the convergence of Fourier series due to the more complicated sign pattern of the terms of the series in comparison to power series. In the present work, the Levin-type H transformation [H.H.H. Homeier, A Levin–type algorithm for accelerating the convergence of Fourier series, Numer. Algo. 3 (1992) 245–254] is studied that involves a frequency parameter α. In particular, properties of the H transformation are derived, and its implementation is discussed. We also present some generalization of it to the case of several frequency parameters. Finally, it is shown how to use the H transformation properly in the vicinity of singularities of the Fourier series. 1991 Mathematics Subject Classification: Primary 65B05, 65B10; Secondary 40A05 40A25 42C15.