Sharpening the Tajkov lower bound in the inequality between the \(C\)- and \(L\)-norms for trigonometric polynomials. (Q1889513)

Given a trigonometric polynomial \(t_n(x)=\sum\limits_{k=-n}^n\hat{t}_n(k)e^{2\pi ikt}\), \newline let \(C_n=\sup\limits_{t_n}\| T_n\| _C/\| t_n\| _L\), where \[ \| t_n\| _C=\max_{| x| \leq1/2}| t_n(x)| \quad\text{and}\quad \| t_n\| _L=\int_{-1/2}^{1/2}| t_n(x)| dx. \] The exact value of \(C_n\) \((n\geq 3)\) is not known. In this paper, the author improves the lower bound for \(C_n\) obtained by \textit{L. V.Taikov} [Usp. Mat. Nauk 20 (3), 205--211 (1965; Zbl 0141.25301)].