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. 
@article{MZM_1998_63_3_a7,
     author = {S. A. Kirillov},
     title = {On a theorem of {Marcinkiewicz} and {Zygmund}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {386--390},
     publisher = {mathdoc},
     volume = {63},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_3_a7/}
}
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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/