Shin-ichi Izumi

Hausdorff means and Gibbs phenomenon

The corresponding phenomenon occurs on the left side of the origin. This is called the Gibbs phenomenon (see [1] and [11]). This phenomenon was discussed for many kinds of means of s~; for example, Ces/tro mean, H61der mean, Euler mean and so on. We shall here consider the Hausdorff mean which contains above all as particular cases. This is defined as follows. Let ~ c, be a given infinite series and s,, be its n-th partial sum. The n-th n = l Hausdorff mean with respect to dg(t) of the series is defined by

Divergence of Fourier series: Corrigenda. II

The proof of divergence of Fourier series which was constructed in the authors earlier paper [ Bull. Austral. Math. Soc. 8 (1973), 289–304] is not complete. The object of this paper is to show that the unestimated terms do not disturb the divergence.