On the absolute summability of trigonometric series (Q1082558)

For an orthonormal system \(\{\phi_ n\}\), consider an orthogonal series \(\sum a_ n\phi_ n\), let \(s_ n\) be its nth partial sum and \(\sigma_ n\) the n-th (C,1) mean. \textit{K. Tandori} [Anal. Math. 6, 157- 164 (1980; Zbl 0453.42019)] showed by counter examples, the statements \((i)\quad \sum | s_{2^{n+1}}-s_{2^ n}| <\infty,\) a.e. and \((ii)\quad \sum | \sigma_{n+1}-\sigma_ n| <\infty,\) a.e. are not necessarily equivalent even if \(\sum a^ 2_ n<\infty\). Considering the behaviors of the above two series for the cosine system, the author proves (I) If \(\{a_ n\}\) is a positive monotone null sequence, then (i) and (ii) are equivalent to \(\sum a_ n/n<\infty\). (II) if \(\{a_ n\}\) is lacunary, i.e., \(a_ j=0\) for \(j\neq n_ k\), \(k_{n+1}/n_ k\geq q>1\) \((k=1,2,...)\), then (i) and (ii) are equivalent to \(\sum | a_ n| <\infty\), and some related results.