On a sequence of trigonometric polynomials (Q1274073)

The following theorem is proved. There exist an absolute constant \(A\) and a sequence of trigonometric polynomials \[ S_n(t):= \sum^n_{k=1} \delta_k(t),\quad n= 1,2,\dots, \] where the \[ \delta_k(t): \sum_{2^k\leq| j|< 2^{k+1}} c_je^{ijt} \] are such that \[ \pi/4\leq\| \delta_k\|_1,\quad \|\delta_k\|_\infty\leq 6,\quad k= 1,2,\dots, n, \] and \[ \| S_n\|_\infty\leq A\sqrt n,\quad n= 1,2,\dots\;. \] {}.