On a proof of K. Tandori (Q2722497)

This paper supplies a detailed proof of key chaining arguments (not appearing in the original article) of the proof by \textit{K. Tandori} [Acta Sci. Math. 26, 249-251 (1965; Zbl 0136.05001)] of the following proposition of \textit{G. Alexits} [``Convergence problems of orthogonal series'' (1961; Zbl 0098.27403)]. Let \(a_n\) be a sequence of positive reals, and \(\varphi_n\) an orthonormal system on \(L^2(X,\mu)\). Then NEWLINE\[NEWLINE\Bigl\|\max_{1\leq m\leq M}|a_1\varphi_1(x)+\ldots+a_m\varphi_m(x)|\Bigr\|^2_{2,\mu} \leq C(\log M)^2 \sum_{k=1}^{M}a_k^2.NEWLINE\]NEWLINE The interest in the chaining argument lies in its use of a weighted binary decomposition of \([1,M]\) using the values \(a_n\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].