On the summability of orthonormal series by the logarithmic methods (Q1113398)

The author introduces iterated logarithmic summability methods. In the first part of the paper their properties are investigated in connection with summation of numerical series. They turn out to be permanent but the different methods non-equivalent. Despite of this, these different methods when applied to orthogonal series with \(\ell^ 2\) coefficient sequences are (a.e.) equivalent. In particular, by a result of J. Meder \(\{\) (log log log n)\({}^ 2\}\) is a sequence of multipliers that ensure a.e. summability. The final part of the work treats analogous questions for strong means.