Geodesic
On a theorem of Marcinkiewicz and Zygmund
Matematičeskie zametki, Tome 63 (1998) no. 3, pp. 386-390.

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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. 
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S. A. Kirillov. On a theorem of Marcinkiewicz and Zygmund. Matematičeskie zametki, Tome 63 (1998) no. 3, pp. 386-390. http://geodesic.mathdoc.fr/item/MZM_1998_63_3_a7/

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