On a theorem of Marcinkiewicz and Zygmund
The author proves the following interesting theorem. There exists an orthogonal system of functions \(\{\varphi_n(x)\}\), \(\varphi_n\in L_\infty[0,1]\) such that for any \(q>2\) there is a sequence \(\{c_n\}\) for which \[ \sum^\infty_{n=1}| c_n|^qn^{q-2}\|\varphi_n\|_\infty^{q-2}<\infty, \] the series \(\sum c_n\varphi_n(x)\) is convergent in \(L_2[0,1]\), but its sum does not belong to \(L_q[0,1]\). This result shows that the classical Marcinkiewicz-Zygmund theorem is sharp.
scientific article; zbMATH DE number 1262561