This article is cited in 1 paper

On a theorem of Marcinkiewicz and Zygmund

S. A. Kirillov

Odessa State Academy of Food Technology

Abstract: 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.

UDC: 517.518

Received: 19.04.1996
Revised: 15.10.1997

DOI: 10.4213/mzm1293

 English version: Mathematical Notes, 1998, 63:3, 338–341

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