Generalization of a theorem of Marcinkiewicz-Zygmund (Q1907406)

Let \(T_n(x)\) be a trigonometric polynomial of order at most \(n\). For the case \(0< p< 1\), the partition \(R= \{z_i\}^\ell_{i= 1}\) \((\ell= \ell(p, n))\) of the segment \([0, 2\pi)\) is found, under which the equivalence of the discrete norm \(|T_n|_{p, R}= \{{1\over \ell} \sum^\ell_{i= 1} |T_n(z_i)|^p\}^{1/p}\) and of the continuous norm \(|T_n|_p= \{\int^{2\pi}_0 |T_n(x)|^p dx\}^{1/p}\) is established. Such a double-sided inequality in the case \(1\leq p< \infty\) is a well-known assertion of Marcinkiewicz-Zygmund [\textit{A. Zygmund}, Trigonometric series, Vol. 1/2, 2nd ed. (1959; Zbl 0085.05601)].