On a theorem of Marcinkiewicz and Zygmund

Let $\{\varphi_n(x)\}$ be an orthonormal system on the closed interval $[0,1]$, and let $\|\varphi_n\|_\infty\le M_n$. In 1937 Marcinkiewicz and Zygmund obtained an estimate of the norm in $L_q[0,1]$ of the sum of the series $\sum_{n=1}^\infty c_n\varphi_n(x)$ under the condition that $\{M_n\}$ is monotone increasing. In this paper it is shown that this condition cannot be discarded.