On a theorem of Marcinkiewicz and Zygmund

Let {ϕn(x)} be an orthonormal system on the closed interval 0,1], and let ∥ϕ _n ∥_∞ ≤M _n . In 1937 Marcinkiewicz and Zygmund obtained an estimate of the norm inL _q 0,1] of the sum of the series ∑ _n=1 ^∞ c _n ϕ _n (x) under the condition that {M _n } is monotone increasing. In this paper it is shown that this condition cannot be discarded. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 386–390, March, 1998. The author wishes to thank V. I. Kolyada for setting the problem and for his permanent attention to this work.